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Problem solution · Java

Minimum Incompatibility

Minimum Incompatibility: a Java solution using dynamic programming. Learn the idea, check the complexity, and read the full code, with credit to walkccc LeetCode Solutions.

Technique
Dynamic programming
Source
walkccc LeetCode Solutions
Length
70 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Dynamic programming

For Minimum Incompatibility, the implementation records answers for smaller states and reuses them to build the requested result without repeating work.

  1. Define precisely what one DP state represents.
  2. Establish the base cases before transitions are evaluated.
  3. Process states in dependency order and combine only already-known values.

Code notes

  • 70 lines of Java from the credited upstream file 1681.java.
  • The implementation visibly relies on sequence storage, cached states.
  • 5 loop blocks detected.

Complexity

Multiply the number of reachable states by the work performed for each transition, then include the stored state table in memory usage.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from walkccc LeetCode Solutions by P.-Y. Chen (walkccc) and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeMinimum Incompatibility · JavaJava
Use this to learn the idea, then write your own version.
class Solution {  public int minimumIncompatibility(int[] nums, int k) {    final int MAX_COMPATIBILITY = (16 - 1) * (16 / 2);    final int n = nums.length;    final int subsetSize = n / k;    final int maxMask = 1 << n;    final int[] incompatibilities = getIncompatibilities(nums, subsetSize);    // dp[i] := the minimum possible sum of incompatibilities of the subset    // of numbers represented by the bitmask i    int[] dp = new int[maxMask];    Arrays.fill(dp, MAX_COMPATIBILITY);    dp[0] = 0;     for (int mask = 1; mask < maxMask; ++mask) {      // The number of 1s in `mask` isn't a multiple of `subsetSize`.      if (Integer.bitCount(mask) % subsetSize != 0)        continue;      // https://cp-algorithms.com/algebra/all-submasks.html      for (int submask = mask; submask > 0; submask = (submask - 1) & mask)        if (incompatibilities[submask] != -1) // valid submask          dp[mask] = Math.min(dp[mask], dp[mask - submask] + incompatibilities[submask]);    }     return dp[maxMask - 1] == MAX_COMPATIBILITY ? -1 : dp[maxMask - 1];  }   private static final int MAX_NUM = 16;   // Returns an incompatibilities array where  // * incompatibilities[i] := the incompatibility of the subset of numbers  //   represented by the bitmask i  // * incompatibilities[i] := -1 if the number of 1s in the bitmask i is not  //   `subsetSize`  private int[] getIncompatibilities(int[] nums, int subsetSize) {    final int maxMask = 1 << nums.length;    int[] incompatibilities = new int[maxMask];    Arrays.fill(incompatibilities, -1);    for (int mask = 0; mask < maxMask; ++mask)      if (Integer.bitCount(mask) == subsetSize && isUnique(nums, mask, subsetSize))        incompatibilities[mask] = getIncompatibility(nums, mask);    return incompatibilities;  }   // Returns true if the numbers selected by `mask` are unique.  //  // e.g. If we call isUnique(0b1010, 2, [1, 2, 1, 4]), `used` variable  // will be 0b1, which only has one 1 (less than `subsetSize`). In this case,  // we should return false.  private boolean isUnique(int[] nums, int mask, int subsetSize) {    int used = 0;    for (int i = 0; i < nums.length; ++i)      if ((mask >> i & 1) == 1)        used |= 1 << nums[i];    return Integer.bitCount(used) == subsetSize;  }   // Returns the incompatibility of the selected numbers represented by the  // `mask`.  private int getIncompatibility(int[] nums, int mask) {    int mn = MAX_NUM;    int mx = 0;    for (int i = 0; i < nums.length; ++i)      if ((mask >> i & 1) == 1) {        mx = Math.max(mx, nums[i]);        mn = Math.min(mn, nums[i]);      }    return mx - mn;  }}

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