Thursday, February 12, 2026

BanquetSplit - Problem on Trees and Bipartite Graphs (Medium)

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Problem

Welcome to the Grand Banquet Dilemma! Picture a gathering of folks connected by a web of relationships. This web of relationships is given as a connected tree with 
 vertices depicting people and 
 edges representing relationships (for each edge the corresponding two people know each other).

You're on a mission to throw an unforgettable party, but there's a twist – you've got two banquets, and you want to ensure that no two people who know each other end up in the same banquet.

Now, your task involves answering 
 queries, each presenting a potential new relationship between two individuals, 
 and 
. The burning question: If this new connection happens, can you still invite everyone to the party using the two banquets? If the answer is "
", you must figure out the minimum number of relationships to remove so that everyone gets an invite and the number of ways to do so.

NOTE: the queries are independent of each other. Adding a new relationship doesn't permanently change the relationship web.

Input format

The first line contains a single integer 
, which denotes the number of test cases.
For each test case:
The first line contains 
 denoting the number of people.
The following 
 lines contain 2 space-separated integers, 
 & 
 indicating that there is a relationship between person person 
 & person 
 
The next line contains 
 denoting the number of queries.
The next 
 lines contain 2 unique space-separated integers, 
 and 
. It's guaranteed that 
 and 
 initially don't have a relationship.
Output format

For each test case, answer all the 
  queries. For each query print 
 if it's possible to invite everyone to the party using the 2 banquets, after adding the current query's relationship to the initial existing 
 relations, ensuring that no two people who know each other end up in the same banquet, otherwise print 
. If the answer is 
, in the next line, print 2 integers, the minimum number of relationships to remove so that everyone gets an invite, and the number of ways to do so.

Constraints


 

Sample Input
1
5
1 2
1 3
3 4
3 5
2
2 4
1 5
Sample Output
Yes
No
1 3
Time Limit: 3
Memory Limit: 256
Source Limit:
Explanation
The first line denotes T = 1.

For test case 
:

We are given:

N = 5
We need to answer 2 queries.
Now,

For the first query, we add a relationship between persons 2 and 4. We can invite everyone by asking persons 1, 4 & 5 to join Banquet 1 and persons 2 & 3 to join Banquet 2, thus ensuring that no two people who know each other end up in the same banquet.
Hence, the answer to the first query is 
.
For the Second query, we add a relationship between persons 1 and 5, and it can be proved that after adding this relationship, we can not invite everyone such that no two people who know each other end up at the same banquet. But,
If we remove the relationship existing between persons 1 and 3, we can invite everyone by asking persons 1 & 3 to join Banquet 1 and persons 2, 5 & 4 to join Banquet 2.
Similarly, If we remove the relationship existing between persons 3 and 5, we can invite everyone by asking persons 1 & 4 to join Banquet 1 and persons 2, 5 & 3 to join Banquet 2.
Similarly, If we remove the relationship existing between persons 1 and 5, we can invite everyone by asking persons 2 & 3 to join Banquet 1 and persons 1, 5 & 4 to join Banquet 2.
Hence, the minimum number of relationships to remove so that everyone gets an invite is 1, and the number of ways to do so is 3.
Hence, the answer to the Second query is 
, followed by 2 integers in the next line as 
 &
Solve it on Hacker Earth

Code (From ChatGPT 5.2) - Passed all tests on HackerEarth

import sys
sys.setrecursionlimit(10**7)
input = sys.stdin.readline

def solve():
    T = int(input())
    for _ in range(T):
        N = int(input())
        
        # Build adjacency list
        adj = [[] for _ in range(N + 1)]
        for _ in range(N - 1):
            u, v = map(int, input().split())
            adj[u].append(v)
            adj[v].append(u)
        
        # Precompute depth and parent table
        LOG = (N).bit_length()
        parent = [[0] * (N + 1) for _ in range(LOG)]
        depth = [0] * (N + 1)
        
        # DFS to compute depth and immediate parent
        def dfs(u, p):
            parent[0][u] = p
            for v in adj[u]:
                if v != p:
                    depth[v] = depth[u] + 1
                    dfs(v, u)
        
        dfs(1, 0)
        
        # Build binary lifting table
        for i in range(1, LOG):
            for node in range(1, N + 1):
                parent[i][node] = parent[i - 1][parent[i - 1][node]]
        
        # LCA function
        def lca(u, v):
            if depth[u] < depth[v]:
                u, v = v, u
            
            # Lift u up to same depth as v
            diff = depth[u] - depth[v]
            for i in range(LOG):
                if diff & (1 << i):
                    u = parent[i][u]
            
            if u == v:
                return u
            
            # Lift both up until parents differ
            for i in reversed(range(LOG)):
                if parent[i][u] != parent[i][v]:
                    u = parent[i][u]
                    v = parent[i][v]
            
            return parent[0][u]
        
        Q = int(input())
        
        for _ in range(Q):
            X, Y = map(int, input().split())
            
            L = lca(X, Y)
            dist = depth[X] + depth[Y] - 2 * depth[L]
            
            # If distance is odd → cycle even → bipartite
            if dist % 2 == 1:
                print("Yes")
            else:
                cycle_length = dist + 1
                print("No")
                print(1, cycle_length)

# Run
solve()

Help From ChatGPT 5.2

I am learning Data Structures and Algorithms.

I will provide a screenshot of a problem statement. 

Your task is NOT to directly give me the final code solution unless I explicitly ask for it.

Instead, please do the following:

1. First, carefully restate the problem in your own words to confirm correct understanding.
   - If anything is unclear in the screenshot, ask me clarification questions before proceeding.

2. Identify:
   - What category of problem this is (e.g., prefix sum, greedy, DP, two pointers, binary search, graph, etc.)
   - What core DSA concepts are being tested
   - What mathematical observations (if any) are involved
   - What constraints are important and how they affect the approach

3. Explain:
   - What naive/brute-force solution would look like
   - Why that may or may not work under the given constraints
   - What optimization insight is required

4. Clearly explain the key trick or insight needed to solve it.
   - Why does that trick work?
   - What pattern should I remember for future problems?

5. Provide:
   - The time complexity and space complexity of the optimal approach
   - Why it satisfies the constraints

6. If there are multiple valid approaches, compare them.

7. Only after all conceptual explanation, provide clean and correct Python code.

8. Avoid guessing unclear details.
   - If any part of the screenshot is ambiguous, ask me instead of assuming.

Focus on correctness over speed. Be explicit in reasoning.

Excellent. This is a very strong graph theory problem. Let’s go step by step exactly as you requested.

1️⃣ Restating the Problem (To Confirm Understanding)

We are given:

  • A graph of N people

  • Initially, there are N - 1 edges

  • So the graph is a tree

Each query gives a potential new relationship (edge) between two people X and Y.

For each query (independent of others):

We temporarily add that edge and ask:

Can we still divide people into 2 banquets such that no two connected people are in the same banquet?

That means:

👉 Is the graph bipartite after adding that edge?

If YES → print "Yes"

If NO:

  • Print "No"

  • Then print:

    • Minimum number of edges to remove to make it bipartite

    • Number of ways to remove that many edges

2️⃣ Identify the Category & Concepts

📌 Category

This is a:

Graph Theory – Bipartite Graph + Tree + Cycle + Parity

📌 Core Concepts Being Tested

  • Tree properties

  • Bipartite graphs

  • Odd cycle detection

  • LCA / depth parity

  • Counting edges in a cycle

  • Graph coloring

  • Fundamental cycle in a tree

📌 Important Observations

  • Initially, graph has N-1 edges → it is a tree

  • A tree is always bipartite.

  • Adding one edge to a tree creates exactly one cycle.

So each query:

  • Adds one edge

  • Creates exactly one cycle

Key idea:

The graph remains bipartite iff that cycle is even length

3️⃣ Naive / Brute Force Thinking

For each query:

  1. Add edge

  2. Try BFS coloring

  3. If conflict → not bipartite

  4. If not bipartite:

    • Try removing each edge one by one

    • Check bipartite again

    • Count ways

Why this fails?

Constraints:

  • N, Q up to 10⁶ across test cases

  • BFS per query → O(N)

  • Q BFS runs → O(NQ) → impossible

So we need something smarter.

4️⃣ The Key Insight (The Big Trick)

🔥 Important fact:

A tree is bipartite.

Adding an edge creates one cycle.

Graph is bipartite iff that cycle is EVEN.

How to check if cycle is even?

Cycle length =
distance(X, Y) in tree + 1

If:

css
distance(X, Y) is even
→ cycle length odd
→ NOT bipartite

If:

css
distance(X, Y) is odd
→ cycle length even
→ bipartite

🔥 EVEN MORE IMPORTANT:

Instead of computing full distance,
just compare depth parity.

If:

pgsql
depth[X] % 2 == depth[Y] % 2
→ distance is even
→ odd cycle
→ NOT bipartite

If depths differ parity:
→ safe.

5️⃣ What If It’s NOT Bipartite?

Then the graph contains exactly one odd cycle.

To fix it:

We must remove at least 1 edge.

Minimum edges to remove = 1

Which edges?

Any edge in that cycle.

How many such edges?

The number of edges in that cycle.

Cycle size =
distance(X, Y) + 1

So:

Minimum removals = 1
Ways = length of cycle

6️⃣ What This Problem Is Teaching

🔥 Core Learning Points

  1. Tree + one extra edge → exactly one cycle

  2. Bipartite graph ↔ no odd cycle

  3. Parity trick using depth

  4. Graph coloring equivalence

  5. Use tree properties instead of brute force

7️⃣ What Design Pattern Is Used?

This is:

Structural property exploitation

Instead of simulating the whole graph each query, we use:

  • Preprocessing tree

  • Depth calculation

  • Parity reasoning

8️⃣ Optimal Approach Outline

Preprocessing (once per test case):

  1. Build adjacency list

  2. Run BFS or DFS from root

  3. Compute:

    • depth[]

    • parent[]

    • maybe binary lifting for LCA (if distance needed)

For each query:

  1. If depth parity differs:
    → print Yes

  2. Else:

    • Find distance between X and Y

    • Cycle length = dist + 1

    • Print:

      • No

      • 1 and cycle_length

9️⃣ Time & Space Complexity

Preprocessing:
O(N)

Each query:

  • Parity check: O(1)

  • Distance using LCA: O(log N)

Total:
O(N + Q log N)

Within constraints.

1️⃣0️⃣ Multiple Approaches?

Yes.

Approach 1:

  • Use LCA to compute exact distance.

Approach 2:

  • Store subtree sizes and depth.

  • Only parity check + precomputed LCA.

Approach 2 is clean and optimal.

1️⃣1️⃣ Why This Is a Strong Problem

It tests:

  • Tree fundamentals

  • Cycle formation logic

  • Bipartite graph property

  • Mathematical reasoning

  • Query handling efficiently

  • Avoiding recomputation

This is typical Codeforces / advanced CP level.

1️⃣2️⃣ Before Giving Code

Let me ask:

Do you want:

  • 🔹 Only conceptual clarity?

  • 🔹 Step-by-step dry run of sample?

  • 🔹 Full Python implementation with LCA?

  • 🔹 Or a simplified competitive programming version?

Tell me how deep you want to go.

User: Full Python implementation with LCA

Excellent — now we’ll build the full correct Python implementation with LCA, step by step, cleanly and efficiently.

This version:

  • Preprocesses the tree once

  • Uses Binary Lifting for LCA

  • Computes distance in O(log N)

  • Answers each query in O(log N)

  • Handles up to 10⁶ nodes across tests

🔥 Key Formula Recap

For query (X, Y):

  1. Compute:

distance(X,Y)=depth[X]+depth[Y]−2∗depth[LCA(X,Y)]distance(X, Y) = depth[X] + depth[Y] - 2 * depth[LCA(X, Y)]

  1. Cycle length = distance + 1

  2. If distance is odd:
    → cycle length even → bipartite → print "Yes"

  3. If distance is even:
    → cycle length odd → not bipartite
    → minimum removals = 1
    → number of ways = cycle_length

✅ Full Python Code (Optimized)

python
import sys
sys.setrecursionlimit(10**7)
input = sys.stdin.readline

def solve():
    T = int(input())
    for _ in range(T):
        N = int(input())
        
        # Build adjacency list
        adj = [[] for _ in range(N + 1)]
        for _ in range(N - 1):
            u, v = map(int, input().split())
            adj[u].append(v)
            adj[v].append(u)
        
        # Precompute depth and parent table
        LOG = (N).bit_length()
        parent = [[0] * (N + 1) for _ in range(LOG)]
        depth = [0] * (N + 1)
        
        # DFS to compute depth and immediate parent
        def dfs(u, p):
            parent[0][u] = p
            for v in adj[u]:
                if v != p:
                    depth[v] = depth[u] + 1
                    dfs(v, u)
        
        dfs(1, 0)
        
        # Build binary lifting table
        for i in range(1, LOG):
            for node in range(1, N + 1):
                parent[i][node] = parent[i - 1][parent[i - 1][node]]
        
        # LCA function
        def lca(u, v):
            if depth[u] < depth[v]:
                u, v = v, u
            
            # Lift u up to same depth as v
            diff = depth[u] - depth[v]
            for i in range(LOG):
                if diff & (1 << i):
                    u = parent[i][u]
            
            if u == v:
                return u
            
            # Lift both up until parents differ
            for i in reversed(range(LOG)):
                if parent[i][u] != parent[i][v]:
                    u = parent[i][u]
                    v = parent[i][v]
            
            return parent[0][u]
        
        Q = int(input())
        
        for _ in range(Q):
            X, Y = map(int, input().split())
            
            L = lca(X, Y)
            dist = depth[X] + depth[Y] - 2 * depth[L]
            
            # If distance is odd → cycle even → bipartite
            if dist % 2 == 1:
                print("Yes")
            else:
                cycle_length = dist + 1
                print("No")
                print(1, cycle_length)

# Run
solve()

🧠 Why This Is Correct

  • Tree + 1 edge → exactly one cycle

  • Bipartite iff cycle is even

  • Cycle length = distance + 1

  • Distance computed using LCA

  • Binary lifting gives O(log N) per query

📊 Complexity

Preprocessing per test case:

mathematica
O(N log N)

Each query:

scss
O(log N)

Total:

mathematica
O(N log N + Q log N)

Works within constraints.

💡 What This Implementation Teaches

  • Binary lifting technique

  • Tree preprocessing

  • Depth-based reasoning

  • Cycle parity trick

  • Avoiding recomputation per query

If you’d like, I can next:

  • 🔎 Walk through sample input step-by-step

  • 📈 Draw a visual diagram explanation

  • 🧠 Explain LCA in deeper detail

  • ⚡ Show how to reduce memory further

  • 🧪 Provide test case stress checker

Just tell me what level of depth you want.

Help From Gemini to Get Concept Clarity On Pointers ChatGPT Threw

...
# Build adjacency list
adj = [[] for _ in range(N + 1)]
for _ in range(N - 1):
    u, v = map(int, input().split())
    adj[u].append(v)
    adj[v].append(u)
...


Q: Why is the adjacency list built using range(N + 1) instead of range(N)?

A:
The adjacency list is built using `range(N + 1)` instead of `range(N)` because the nodes in the graph are likely numbered from 1 to N, rather than from 0 to N-1. By using `range(N + 1)`, we create an adjacency list that can accommodate all node indices up to N, including the index 0 which may not be used. This allows for easier indexing when adding edges between nodes, as we can directly use the node numbers without needing to adjust for zero-based indexing.

The statement "A graph of N people with N-1 edges is a tree" is a famous rule of thumb in graph theory (the math of connections), but it relies on a specific "sweet spot" of efficiency.

Here is the layman explanation of how that conclusion is reached.

1. The Analogy: Islands and Bridges

Imagine the people are Islands, and the edges are Bridges.

  • Goal: You want to make it possible to walk from any island to any other island (the graph is "connected").

  • Constraint: You want to build the absolute minimum number of bridges possible to save money.

How many bridges do you need?

  • To connect 2 islands, you need 1 bridge.

  • To connect 3 islands, you need 2 bridges.

  • To connect 4 islands, you need 3 bridges.

  • To connect N islands, you need N - 1 bridges.

If you have exactly bridges and the islands are all connected, you have built a perfect structure with no loops and no waste. This structure is called a Tree.

2. Why is it called a "Tree"?

In math, a "Tree" has two main rules:

  1. Everything is connected: You can reach any node from any other node.

  2. No Loops (Cycles): There are no redundant paths. If you can get from A to B, there is only one way to do it.

The Logic of N-1:

  • If you had fewer than edges: It would be impossible to connect everyone. Someone would be left stranded on an isolated island.

  • If you had more than edges: You would accidentally create a loop (a cycle). You would have a "redundant" bridge that connects two places that were already reachable.

3. The "Hidden" Assumption

There is a tiny catch. The statement you quoted usually implies that the graph is connected.

If the graph is not connected, the conclusion is false.

  • Example: Imagine 4 people (). You have 3 edges ().

  • You could form a triangle with 3 people (using 3 edges) and leave the 4th person completely alone.

  • This has edges, but it has a loop (the triangle) and is not connected. It is not a tree.

Summary: The text you are reading concludes it is a tree because it assumes the network is efficiently connected. It assumes that edges are used perfectly to link everyone together without creating any loops, which creates the "Tree" structure.

Would you like me to explain how this concept is used in computer algorithms like "Minimum Spanning Trees"?

In layman terms, a Bipartite Graph is a fancy mathematical way of saying: "Can we split everyone into two opposing teams without any internal conflicts?"

Since your problem mentions "2 banquets," here is the exact analogy:

1. The "Two Banquet Halls" Rule

Imagine you are organizing a dinner party and you have exactly two banquet halls (let's call them Hall A and Hall B).

  • The Rule: If two people know each other (are connected by an "edge"), they hate each other. They cannot sit in the same hall. They must be separated.

  • The Goal: You need to assign every single person to either Hall A or Hall B so that no two enemies end up in the same room.

If you can successfully separate everyone like this, the graph is Bipartite.

2. When does it fail? (The "No" Case)

The graph stops being bipartite when you have a conflict that forces you to break the rule. This usually happens when you have a "Triangle" (or any loop with an odd number of people).

The Triangle Example: Imagine Alice, Bob, and Charlie all know each other.

  1. Alice goes to Hall A.

  2. Bob knows Alice, so he must go to Hall B.

  3. Charlie knows Bob, so he should go to Hall A.

  4. BUT: Charlie also knows Alice (who is already in Hall A).

Charlie has nowhere to go. He conflicts with someone in Hall A and someone in Hall B.

  • Because this group of 3 cannot be split into 2 rooms, the graph is not bipartite.

3. Connecting it to your Code Problem

Your problem is asking you to simulate this party planning process:

  1. "Temporarily add that edge": You introduce a new rivalry/connection between two people.

  2. "Is the graph bipartite?": You check: "Did this new connection create an impossible triangle (or odd loop) that breaks our seating plan?"

    • YES: The seating plan still works. Everyone is happy.

    • NO: The new connection created a conflict. You now have an odd cycle (like the triangle above).

Why remove edges? If the answer is "No," the problem asks: "Okay, the plan is broken. What is the minimum number of connections we need to ignore (delete) to fix the seating plan?"

In the Triangle example (Alice-Bob-Charlie), you would need to remove 1 edge (e.g., ignore the fact that Alice knows Charlie) to make the seating plan work again.

Would you like me to explain the algorithm (BFS/DFS Coloring) usually used to check this?

You said

why does the literature use the terms like the triangle or odd cycle Would it be possible to put 4 people who know each other into two groups where no people know each other?

The literature uses the term "Odd Cycle" because odd numbers are the natural enemy of "even" splits. It all comes down to the math of alternating.

1. Why "Odd Cycles" and "Triangles"?

In graph theory, a "cycle" is just a path that starts and ends at the same person, forming a loop.

  • Even Cycles (2, 4, 6 people): These work perfectly. You can alternate: Hall A, Hall B, Hall A, Hall B. Because the number is even, the last person (Hall B) will connect back to the first person (Hall A) without a conflict.

  • Odd Cycles (3, 5, 7 people): These always fail. If you try to alternate A-B-A-B-A, the 5th person is in Hall A, but they are also connected back to the 1st person... who is also in Hall A.

The Triangle is simply the smallest possible odd cycle (3 people). It is the most basic "building block" of a non-bipartite graph. If your graph contains even one triangle (or any odd-sized loop), the whole thing is no longer bipartite.

2. Can 4 people who know "each other" be split?

It depends on how they know each other. In graph terms: "Who is connected to whom?"

Case A: The "Square" (Bipartite)

If the 4 people know each other in a simple loop (A knows B, B knows C, C knows D, and D knows A), you can split them.

  • Hall A: Person 1 and Person 3

  • Hall B: Person 2 and Person 4 Since 1 doesn't know 3, and 2 doesn't know 4, there are no conflicts!

Case B: The "Complete Graph" (NOT Bipartite)

If you mean "everyone knows everyone else" (Person 1 knows 2, 3, and 4; Person 2 knows 1, 3, and 4, etc.), then no, it is impossible.

  • If you put Person 1 in Hall A, then 2, 3, and 4 must all go to Hall B because they all know Person 1.

  • But wait! You said everyone knows everyone. That means 2, 3, and 4 all know each other.

  • Putting 2, 3, and 4 in Hall B together creates massive conflicts.

The Rule of Thumb: If you have a "Clique" (a group where everyone knows everyone) of 3 or more people, you can never split them into just two groups. To split a 4-person "everyone-knows-everyone" group, you would actually need 4 separate rooms!

Summary

  • Odd Cycle: A loop of people that is impossible to split into two (3, 5, 7...).

  • Triangle: The simplest "illegal" shape for a 2-banquet setup.

  • 4 People: Can be split if they form an even loop (a square), but cannot be split if they all know each other (a clique).

Would you like to see how we use "Graph Coloring" (using just two colors) to prove if a graph is bipartite?

You said

Is this true "A tree is always bipartite."

Can you explain in layman terms?

Yes, that is absolutely true. Every tree, no matter how large or complicated it looks, is always bipartite.

In layman’s terms, this means you can always split a tree into two groups (Banquets A and B) without any conflicts. Here is why:

1. The "Generational" Logic

Think of a tree like a literal family tree (but without the messy "everybody knows everyone" complexity).

  • Generation 1: Pick one person to be the "Root" and put them in Hall A.

  • Generation 2: All their direct connections (their "children") must go to Hall B.

  • Generation 3: All of their connections (the "grandchildren") must go back to Hall A.

  • Generation 4: All the "great-grandchildren" go to Hall B.

Because it is a tree, there are no loops. This means a "grandchild" can never suddenly be connected back to the "root" or to their own parent's sibling. They only connect to the generation directly above or below them.

2. Why it Works (No "Shortcuts")

Remember, for a graph to fail at being bipartite, it needs an Odd Cycle (like a triangle).

In a tree, there are no cycles at all. There is only one unique path between any two people. To create a conflict, you would need a "shortcut" that connects two people in Hall A together. But if you added a shortcut like that, you would create a loop—and the moment you have a loop, it’s not a tree anymore.

3. The "Alternating Colors" Rule

If you imagine walking through a tree:

  1. Start at any person.

  2. Every time you move along an edge to a new person, you switch groups (Hall A Hall B Hall A).

  3. Since there are no loops, you will never arrive at a person and find out they were already assigned to the "wrong" hall.

Summary

A tree is the ultimate "conflict-free" structure for two banquets. Because it lacks cycles, you can simply alternate halls level-by-level, and you will never run into a situation where two people in the same hall are connected.

Since we know the starting graph in your problem is a tree, it starts off as bipartite. Would you like to know how adding a single edge can "break" that tree and turn it into a non-bipartite graph?

To understand this, let’s go back to our Two Banquet Halls analogy. Remember: as long as the graph is a tree, everyone is perfectly split between Hall A and Hall B.

The trouble starts when we add a new relationship (edge) between two people, X and Y, who weren't connected before. This "shortcut" creates a Cycle.

1. What is Distance(X, Y)?

In a tree, the distance is simply the number of steps (edges) you have to take to get from person X to person Y using the existing connections.

  • Distance 1: They are direct friends.

  • Distance 2: They have one mutual friend between them.

  • Distance 3: There is a "friend of a friend of a friend" chain.

2. What is Cycle Length?

When you add a new direct edge between X and Y, you close a loop. The Cycle Length is the total number of people involved in that loop.

  • The Formula:

  • Why the "+1"? The "Distance" is the old path around the long way. The "+1" is the new shortcut you just added. Together, they form a complete circle.

3. Why the Odd/Even rule works

This is the "aha!" moment. Since the tree is already split into Hall A and Hall B, everyone is sitting in a hall based on their distance from the "Root."

Case A: Distance is Even (Bad News ❌)

If the distance between X and Y is even (like 2 steps), it means X and Y are currently sitting in the same hall.

  • Example: X is in Hall A. Their friend is in Hall B. Their friend's friend (Y) is back in Hall A.

  • The Conflict: If you suddenly add a direct edge between X and Y, you are connecting two people who are both in Hall A.

  • The Cycle: (A Triangle!). 3 is an odd number.

  • Result: NOT Bipartite.

Case B: Distance is Odd (Good News ✅)

If the distance between X and Y is odd (like 3 steps), it means X and Y are currently in different halls.

  • Example: X is in Hall A Friend 1 (B) Friend 2 (A) Y (B).

  • The Logic: Since X is in Hall A and Y is in Hall B, adding a direct edge between them is perfectly fine! They are already in separate rooms.

  • The Cycle: (A Square). 4 is an even number.

  • Result: Still Bipartite.

Summary Table

Distance between X and YThey are currently in...New Cycle LengthIs it okay?
Even (2, 4, 6...)The Same HallOdd (3, 5, 7...)No (Conflict!)
Odd (1, 3, 5...)Different HallsEven (2, 4, 6...)Yes (Safe!)

Would you like to know how programmers quickly find the "Distance(X, Y)" in a tree without counting every single person by hand?

To understand Depth Parity, think of our tree as an office building with many floors.

1. What is Depth?

The Depth of a person is simply which "floor" they are sitting on, starting from the boss (the Root) at the very top.

  • Depth 0 (Root): The Boss.

  • Depth 1: The Boss's direct assistants.

  • Depth 2: The assistants' employees.

2. What is Parity?

Parity is just a fancy word for whether a number is Even or Odd.

  • Even Parity: Floors 0, 2, 4, 6...

  • Odd Parity: Floors 1, 3, 5, 7...

3. The Banquet Connection

In a tree, we assign people to halls based on their floor:

  • Everyone on Even floors goes to Hall A.

  • Everyone on Odd floors goes to Hall B.

Because it's a tree, an even-floor person only connects to odd-floor people (their "boss" above or "employee" below). This keeps the halls perfectly separated.

4. Why compare "Depth Parity"?

When the computer asks: depth[X] % 2 == depth[Y] % 2, it is really asking: "Are these two people currently assigned to the same banquet hall?"

If Depth Parity is the SAME (e.g., both are Even)

  • Situation: Both X and Y are already sitting in Hall A.

  • Conflict: If you add a relationship (edge) between them, you are connecting two people in the same room.

  • Result: This creates an Odd Cycle (like a triangle or a 5-person loop). The graph is NOT Bipartite.

If Depth Parity is DIFFERENT (e.g., one is Even, one is Odd)

  • Situation: X is in Hall A and Y is in Hall B.

  • Harmony: If you add an edge between them, it’s perfectly fine! They are already in different rooms, so the "no enemies in the same room" rule is still followed.

  • Result: The cycle created is Even (like a square). The graph is STILL Bipartite.

Summary in one sentence

Comparing depth parity is a shortcut to see if two people are already in the same banquet hall; if they are, adding a connection between them breaks the "Bipartite" rule.

Would you like me to explain how "Minimum number of edges to remove" works if you find a conflict?

This formula looks intimidating, but it’s actually a very clever way of "counting steps" on a family tree without having to walk the path manually.

1. What is the LCA?

To find the distance between two people (X and Y), you first have to find their LCA (Lowest Common Ancestor).

  • Think of this as the "Greatest Grandparent" that both X and Y have in common.

  • It is the point where their two branches of the tree finally meet.

2. Breaking Down the Formula

Imagine X is on Floor 5, and Y is on Floor 4. Their Common Ancestor (LCA) is way up on Floor 1.

Why do we add the depths? If you just add (5) and (4), you get 9. This is the total number of steps from the very top of the building down to X, plus the steps from the top down to Y.

Why subtract ? By adding both depths, you have "double-counted" the hallway from the very top (the Root) down to the Common Ancestor.

  1. You counted it once while going down to X.

  2. You counted it again while going down to Y.

To find the distance only between X and Y, you have to "cut off" those two shared top sections.

  • Subtracting once removes X's path to the root.

  • Subtracting it a second time removes Y's path to the root.

Example:

  • X is at depth 5.

  • Y is at depth 4.

  • LCA is at depth 1.

  • steps.

3. The "Minimum Removals" Logic

If the distance is even, we know we have an odd cycle (a conflict). The problem asks: "How do we fix this with the minimum number of removals?"

  • Minimum Removals = 1: To break any single loop, you only need to cut one connection. If you cut one edge of a triangle, it's no longer a loop!

  • Number of ways = Cycle Length: Since the cycle is a perfect loop of people (say, 5 people), any one of the 5 "relationships" (edges) in that loop is responsible for keeping the loop closed. If you remove any one of those 5 edges, the "Odd Cycle" disappears and the graph becomes Bipartite again. So, there are 5 different ways to fix it.

Summary Checklist

  • LCA: The meeting point of two branches.

  • The Formula: Adds the total "height" of both people and removes the "shared" height above their ancestor.

  • The "Ways": If you have a loop of 5 people, you have 5 different edges you could potentially delete to solve the problem.

Would you like to know why "Distance is even" always leads to an "Odd Cycle"?

You said

What is "adjacency list"

In layman's terms, an Adjacency List is like a contact list for a network. It is the most common way for a computer to store a graph (a collection of people and their relationships).

1. The Real-World Analogy

Think of your phone’s contact app. You don't have a giant, complicated map of the whole world. Instead:

  • You have a list of names (Nodes).

  • Under each name, you have a list of people they are connected to (Edges).

Example: A small group of friends

  • Alice knows Bob and Charlie.

  • Bob knows Alice and David.

  • Charlie knows Alice.

  • David knows Bob.

In an Adjacency List, the computer stores it exactly like that:

Alice: [Bob, Charlie]

Bob: [Alice, David]

Charlie: [Alice]

David: [Bob]

2. Why use this instead of a "Grid"?

There is another way to store graphs called an Adjacency Matrix (a giant grid of 1s and 0s). Imagine if you had a grid of every person in the world. To record that Alice knows Bob, you'd need a table with 8 billion rows and 8 billion columns. Most of that table would be empty (zeros) because Alice doesn't know 8 billion people!

The Adjacency List is much smarter because:

  • It saves space: You only store the connections that actually exist. If Alice only has 2 friends, her list only has 2 entries.

  • It's fast for "Who are your friends?": If the computer wants to know who Alice knows, it just looks at her specific list. It doesn't have to scan through a giant empty grid.

3. How it looks in Code

If you are writing a program, an adjacency list is usually just an Array of Lists.

  • The Index of the array represents the person (Person 0, Person 1, Person 2...).

  • The Value at that index is a list containing the IDs of their friends.

Python
# A simple Adjacency List in Python
graph = {
    0: [1, 2],  # Person 0 is connected to 1 and 2
    1: [0, 3],  # Person 1 is connected to 0 and 3
    2: [0],     # Person 2 is connected to 0
    3: [1]      # Person 3 is connected to 1
}

Summary

An Adjacency List is simply a collection of "mini-lists." Each person in your graph has their own small list of neighbors. It is the most efficient way to tell the computer who is connected to whom in a large network.

Would you like to know how the computer uses this list to find the "Distance(X, Y)" we talked about earlier?

This code is the "preparation phase" for a technique called Binary Lifting. Its job is to help the computer find the LCA (the common ancestor) and the distance between two people almost instantly, even if there are millions of people in the tree.

Here is the breakdown in layman terms:

1. LOG = (N).bit_length()

The "Power of Two" Ruler Imagine you are standing at the bottom of a giant staircase. You want to climb up, but instead of taking one step at a time, you want to be able to jump in powers of two ().

  • (N).bit_length() calculates how many "doubling" steps you need to cover the entire height of the tree.

  • If you have 1,000 people, LOG will be 10 (because ). This tells the computer: "The longest jump we will ever need to make is steps."

2. parent = [[0] * (N + 1) for _ in range(LOG)]

The "Fast-Travel" Map This is the most important part. A normal adjacency list only tells you who your immediate "parent" is (1 step up). This parent table is a giant cheat sheet that stores your ancestors at different "jump" heights.

  • parent[0][X] stores who is 1 step () above person X.

  • parent[1][X] stores who is 2 steps () above person X.

  • parent[2][X] stores who is 4 steps () above person X.

  • parent[3][X] stores who is 8 steps () above person X.

Why do this? If the computer needs to move person X up 13 floors to find an ancestor, it doesn't do 13 individual steps. It looks at its cheat sheet and does:

  • One jump of 8 floors.

  • Then a jump of 4 floors.

  • Then a jump of 1 floor.

  • Total: 13 floors in just 3 moves!

3. depth = [0] * (N + 1)

The "Floor Level" Tracker This is a simple list where the computer stores which floor everyone is on.

  • depth[5] = 3 means Person #5 is on the 3rd floor.

As we discussed earlier, we need this "floor number" to calculate the distance formula () and to check the Parity (Even vs. Odd floors).

Summary

This piece of code is basically setting up a High-Speed Elevator System.

  • LOG decides how big the elevator buttons should go.

  • parent builds the "express tracks" so you can jump floors.

  • depth records exactly which floor everyone is standing on.

Would you like to see the next part of the code, where the computer actually fills in these "express tracks" using the immediate parents?

You said

explain this in layman terms:


        # DFS to compute depth and immediate parent

        def dfs(u, p):

            parent[0][u] = p

            for v in adj[u]:

                if v != p:

                    depth[v] = depth[u] + 1

                    dfs(v, u)

This part of the code is the "First Pass" through the network. It’s where the computer actually walks through the tree to map out who everyone's immediate boss is and what floor they are sitting on.

1. What is DFS?

DFS stands for Depth-First Search. In layman terms, this is the "Explorer" approach. Instead of looking at all his neighbors first, the computer picks one branch and follows it all the way to the very bottom (the "leaves") before coming back up and trying a different branch.

2. Breaking Down the Code Lines

parent[0][u] = p

This line is the computer taking a note: "The person 1-step () above u is p." It’s like filling in the "Father" column on a birth certificate. We need this basic info before we can calculate the "Grandfather" and "Great-Grandfather" jumps later.

for v in adj[u]:

The computer looks at the Adjacency List for person u to see everyone they are connected to.

  • u is the current person.

  • v is a potential connection.

if v != p:

This is a safety check. In a tree, connections go both ways (Up to the parent and Down to the children). We don't want the computer to get stuck in a loop going: "I'm looking at my child... now I'm looking back at my parent... now back to the child." This line says: "Only move forward if this person isn't the one who just gave birth to me."

depth[v] = depth[u] + 1

This is how the floor numbers are assigned. If I am on Floor 5 (depth[u]), and you are my child (v), then you must be on Floor 6.

dfs(v, u)

This is the Recursion. The computer now "steps into" the child's shoes and repeats the whole process for that child's own kids.

3. The Big Picture

By the time this function finishes running, every person in the tree has:

  1. A Floor Number: They know exactly how far they are from the Root.

  2. A Direct Parent: They know exactly who is one step above them.

This creates the foundation. Once you have the 1st parent, the computer can use logic to figure out the 2nd parent (the parent's parent), the 4th parent (the 2nd parent's 2nd parent), and so on.

Would you like to see the "Fill" logic that creates those high-speed jumps (2, 4, 8 steps) now that the 1st parent is known?

This is the "magic" part of the code where the computer builds the High-Speed Elevator we talked about earlier.

Instead of walking up the stairs one by one, the computer uses logic to calculate "giant leaps."

1. The Core Idea: "A Jump of a Jump"

The line parent[i][node] = parent[i - 1][parent[i - 1][node]] is a mathematical shortcut. To understand it, think about this:

  • To jump 2 steps up, you jump 1 step, and then jump another 1 step from there.

  • To jump 4 steps up, you jump 2 steps, and then jump another 2 steps from there.

  • To jump 8 steps up, you jump 4 steps, and then jump another 4 steps from there.

2. Breaking Down the Code

for i in range(1, LOG):

This loop decides the size of the jump.

  • When i = 1, we are calculating 2-step jumps ().

  • When i = 2, we are calculating 4-step jumps ().

  • When i = 3, we are calculating 8-step jumps ().

for node in range(1, N + 1):

This goes through every person in the tree, one by one, to fill out their personal "jump cheat sheet."

parent[i][node] = ...

This is where the calculation happens. Let's look at it for i = 2 (the 4-step jump): parent[2][node] = parent[1][ parent[1][node] ]

In plain English, this says:

"To find out who is 4 steps above me, first find out who is 2 steps above me. Then, look at that person's cheat sheet to see who is 2 steps above them."

3. Why is this so powerful?

Because of this "doubling" logic, the computer only needs to do a very small amount of work to prepare for massive trees.

  • If you have 1,000,000 people, the computer only needs to run this loop about 20 times (LOG would be 20).

  • Once this table is built, the computer can find an ancestor 500,000 levels up in just a few microseconds by combining these jumps (e.g., a 262,144 jump + a 131,072 jump, etc.).

Summary

This code is pre-calculating the "grandparents," "great-grandparents," and "great-great-great-grandparents" for every single person. It uses the work it did for smaller jumps to instantly calculate larger ones.

Now that the elevator is built, would you like to see the code for the "LCA" function—the one that actually uses these jumps to find where two people's branches meet?

Finding the LCA (Lowest Common Ancestor) is like finding the first common relative between two people on a family tree. This function uses the "High-Speed Elevator" (Binary Lifting table) we just built to do it in record time.

It happens in three main steps:

Step 1: Making the depths equal

Imagine X is on the 10th floor and Y is on the 4th floor. It’s hard to find their meeting point when they are at different heights.

Python
if depth[u] < depth[v]:
    u, v = v, u  # Make sure 'u' is the deeper one

First, the code ensures u is the one further down. Then, it "lifts" u up until it is on the same floor as v.

Python
# Lift u up to same depth as v
diff = depth[u] - depth[v]
for i in range(LOG):
    if diff & (1 << i):
        u = parent[i][u]

If the difference is 6 floors, the computer doesn't take 6 steps. It looks at the binary version of 6 (which is ). It uses the elevator to jump 4 floors, then 2 floors. Now, u and v are eye-to-eye on the same level.

Step 2: Checking if we already found it

Python
if u == v:
    return u

After lifting u up, we check: "Are you now standing on the exact same spot as v?" If yes, it means v was an ancestor of u all along. We’re done!

Step 3: The "Sync Jump" to the top

If they are on the same floor but in different branches, they start jumping upward together.

Python
# Lift both up until parents differ
for i in reversed(range(LOG)):
    if parent[i][u] != parent[i][v]:
        u = parent[i][u]
        v = parent[i][v]

The computer tries the biggest possible jump (say, 128 floors).

  • If a 128-floor jump lands them on the same person, the jump was too big (they passed the LCA). It tries a smaller jump.

  • If a 64-floor jump lands them on different people, they both take that jump and continue from there.

By trying every jump size from largest to smallest, they land exactly one step below their common ancestor.

Step 4: The Final Result

Python
return parent[0][u]

Since u and v are now standing exactly one level below the common ancestor, the function simply returns their immediate parent (parent[0]). That person is the LCA!

Summary in Layman Terms

  1. Level the playing field: Lift the lower person up so both are on the same floor.

  2. The Chase: If they aren't the same person, make them jump up the tree together using the biggest possible "leaps" that don't make them crash into each other.

  3. The Meeting: Once they can't jump any higher without hitting the same person, their very next step up is their meeting point.

Now that you have the LCA, you can solve the whole "Banquets" problem! Would you like to see how we put all these pieces together to answer the final "Yes" or "No" queries?

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