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#3656MediumPremium on LC~35 min

Determine if a Simple Graph Exists

Time O(nlogn) · Space O(1) · Official statement on LeetCode

graphgraphsortprefix sumtwo pointers
cpppython

Solutions

// Time:  O(nlogn)
// Space: O(1)

// Erdős–Gallai theorem, sort, prefix sum, two pointers
class Solution {
public:
    bool simpleGraphExists(vector<int>& degrees) {
        // reference: https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Gallai_theorem
        const auto& total = accumulate(cbegin(degrees), cend(degrees), 0LL);
        if (total % 2) {
            return false;
        }
        sort(begin(degrees), end(degrees), greater<int>());
        int64_t lhs = 0, suffix1 = total, suffix2 = 0;
        for (int k = 1, i = size(degrees) - 1; k <= size(degrees); ++k) {
            lhs += degrees[k - 1];
            suffix1 -= degrees[k - 1];
            for (; i >= 0 && degrees[i] < k; --i) {
                suffix2 += degrees[i];
            };
            const auto& rhs = static_cast<int64_t>(k) * (k - 1) + ((i - k + 1 > 0) ? static_cast<int64_t>(i - k + 1) * k + suffix2: suffix1);
            if (!(lhs <= rhs)) {
                return false;
            }
        }
        return true;
    }
};

Beginner Explanation

What is Determine if a Simple Graph Exists?

Determine if a Simple Graph Exists (LeetCode #3656) is a Medium problem that primarily trains graph.

How to think about it

  1. Restate the goal in your own words before coding.
  2. Work a tiny example by hand so the invariant becomes obvious.
  3. Identify the pattern — this problem aligns with graph, sort, prefix sum, and two pointers.
  4. Only then translate the idea into code.

Why this problem matters

It sits in the sweet spot of interview difficulty: multiple valid approaches, clear trade-offs. Official solution notes mention: Graph, Erdős–Gallai Theorem, Sort, Prefix sum.

AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.

Interview Walkthrough

Interview approach for Determine if a Simple Graph Exists

Opening (30–60 seconds)

  • Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
  • State a brute force so the interviewer knows you can solve it naively.
  • Propose the optimal direction tied to graph, sort, prefix sum, and two pointers.

Core solution narrative

  1. Define the state you track (pointers, DP cell, set membership, stack top, etc.).
  2. Explain the transition when you process the next element.
  3. Call out time (O(nlogn)) and space (O(1)) before coding.
  4. Code cleanly; narrate variable names.

What interviewers listen for

  • Correctness on edge cases
  • Complexity honesty
  • Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)

Follow-up questions they may ask

  • Can you solve it with less memory?
  • What if the input stream is infinite / doesn't fit in RAM?
  • How would tests look for adversarial inputs?

Optimized Approach

Optimized solution notes

The reference solutions on AlgoForge target O(nlogn) time and O(1) space.

Pattern focus: graph, sort, prefix sum, and two pointers

Use the pattern as a checklist:

  • graph — confirm the invariant holds after each step
  • sort — confirm the invariant holds after each step
  • prefix sum — confirm the invariant holds after each step
  • two pointers — confirm the invariant holds after each step

Start from the primary solution, then rewrite from memory to lock it in.

Implementation tips

  • Prefer readable names over micro-optimizations in interviews.
  • Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
  • After AC-level logic, re-scan for off-by-one and null checks.

Complexity Analysis

Complexity

Measure Bound
Time O(nlogn)
Space O(1)

How to justify this in an interview

  • Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
  • Space: include hash maps, recursion stack, and output allocation when the problem asks for it.

If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.

Common Mistakes

Common mistakes on Determine if a Simple Graph Exists

  1. Skipping edge cases — empty collections, single-element inputs, max constraints.
  2. Wrong invariant for graph, sort, prefix sum, and two pointers — updating state too early or too late.
  3. Mutating input unexpectedly when the problem forbids it.
  4. Off-by-one in windows, ranges, or binary search bounds.
  5. Ignoring overflow / precision for integer arithmetic problems.
  6. Overengineering — jumping to an advanced structure when a simpler approach works.

Alternative Approaches

AI expand later

Alternatives

Placeholder for multi-approach comparison. Future AI content generation can expand:

  • Brute force baseline
  • Optimal graph, sort, prefix sum, and two pointers solution
  • Space-optimized rewrite

Prompt slot: expand alternatives for determine-if-a-simple-graph-exists.

Edge Cases

Edge cases checklist

  • Minimum input size
  • Maximum input size / time limits
  • Duplicates and already-sorted input
  • Negative numbers / zeros (if applicable)
  • Disconnected structures (graphs/trees)
  • Single path vs branching recursion depth

Pattern Recognition

Spotting this pattern

Signal phrases that point to graph, sort, prefix sum, and two pointers:

  • Sorted input or ability to sort without changing the answer class
  • Need for contiguous subarray / substring → consider sliding window
  • Need for O(1) membership → hash set/map
  • Optimal substructure + overlapping subproblems → DP
  • Connectivity / components → graph DFS/BFS or Union-Find

Primary topics: graph.

Follow-up Interview Questions

Follow-ups

  1. How does the solution change if the input is a stream?
  2. Can you solve it in-place?
  3. What if duplicates must be handled differently?
  4. How would you parallelize the approach?
  5. Design tests that would break a buggy implementation.

Practice Recommendations

What to practice next

  1. Re-solve Determine if a Simple Graph Exists in a second language (cpp, python).
  2. Drill 3–5 more problems tagged graph.
  3. Teach the solution out loud in under 5 minutes.
  4. Add this problem to your revision calendar in 3 days and 14 days.

Visualization

Conceptual diagram for Determine if a Simple Graph Exists: show input structure (graph), highlight the moving parts of the graph, sort, prefix sum, and two pointers approach, and annotate each step with the maintained invariant and complexity.

Study checklist

  • Read the official problem statement on LeetCode
  • Solve on paper / whiteboard first
  • Implement the graph, sort, prefix sum, and two pointers approach
  • Verify edge cases from the checklist
  • State time and space complexity aloud
  • Compare with the AlgoForge reference solution
  • Schedule a revision session

Revision notes

Determine if a Simple Graph Exists (#3656) — Medium. Pattern: graph, sort, prefix sum, and two pointers. Complexity: O(nlogn) time / O(1) space. Re-derive the invariant before coding.

FAQs

What is the time complexity of Determine if a Simple Graph Exists?+

The reference solutions aim for O(nlogn) time and O(1) space. Always re-derive complexity from the code you write in the interview.

What pattern does Determine if a Simple Graph Exists use?+

It primarily maps to graph, sort, prefix sum, and two pointers, within the broader topic of graph.

Is Determine if a Simple Graph Exists good for interviews?+

Yes — as a Medium problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.

Where can I read the official statement?+

Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/determine-if-a-simple-graph-exists/