Number of ZigZag Arrays III - Solution & Explanation

Problem Overview: You are given constraints for building arrays that follow a zigzag pattern, meaning adjacent elements must alternate between increasing and decreasing. The task is to count how many valid arrays satisfy this alternating pattern while respecting the value bounds.

Approach 1: Brute Force Enumeration (Exponential Time, O(m^n) time, O(n) space)

The most direct idea is to generate every possible array of length n with values in the allowed range and check whether it forms a valid zigzag pattern. During generation, you track the previous element and verify that the direction alternates between < and >. This uses recursive backtracking to build arrays element by element. While simple to reason about, the search space grows exponentially with m^n, making it infeasible except for tiny inputs.

Approach 2: Dynamic Programming with Direction State (O(n*m^2) time, O(n*m) space)

A better strategy stores partial results. Define dp[i][v][d] as the number of zigzag arrays of length i ending with value v, where d indicates whether the last comparison was increasing or decreasing. For each new position, iterate over all previous values that satisfy the alternating condition. If the previous step was increasing, the next value must be smaller; if decreasing, the next must be larger. This reduces the problem to structured transitions between states, which is a common pattern in dynamic programming. The main cost comes from scanning all possible previous values for each transition.

Approach 3: DP with Prefix Sum Optimization (O(n*m) time, O(n*m) space)

The quadratic transition in the previous method can be optimized. Instead of iterating through all valid previous values each time, maintain prefix sums of DP counts. When you need the number of ways where the previous value is smaller or larger than the current value, compute it in constant time using cumulative sums. This converts the expensive nested loops into simple prefix lookups. The technique is similar to range-sum transitions used in prefix sum optimization for DP. The result is an O(n*m) algorithm that scales to large constraints.

Approach 4: Combinatorial DP Compression (O(n*m) time, O(m) space)

Space can be reduced by observing that each row of the DP depends only on the previous row. Keep two arrays representing counts for increasing and decreasing endings, then update them using prefix sums. This rolling-array technique is common in memory-optimized dynamic programming. The time complexity remains linear in the state space while reducing memory usage significantly.

Recommended for interviews: Start by describing the brute-force generation to show understanding of the zigzag constraint. Then move quickly to the dynamic programming formulation with direction states. The optimized DP using prefix sums is the solution interviewers typically expect because it reduces the transition from O(m) to O(1), bringing total complexity down to O(n*m).