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Problem solution · Python

Minimum Operations to Make Elements Within K Subarrays Equal

Minimum Operations to Make Elements Within K Subarrays Equal: a Python solution using dynamic programming. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Dynamic programming
Source
Kamyu LeetCode Solutions
Length
190 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 Operations to Make Elements Within K Subarrays Equal, 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

  • 190 lines of Python from the credited upstream file minimum-operations-to-make-elements-within-k-subarrays-equal.py.
  • The implementation visibly relies on sequence storage, hash lookup, work queue, cached states.
  • No explicit 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 Kamyu LeetCode Solutions by kamyu104 and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeMinimum Operations to Make Elements Within K Subarrays Equal · PythonPython
Use this to learn the idea, then write your own version.
# Time:  O(nlogx + k * n)# Space: O(n) from sortedcontainers import SortedList  # two sorted lists, dpclass Solution(object):    def minOperations(self, nums, x, k):        """        :type nums: List[int]        :type x: int        :type k: int        :rtype: int        """        class SlidingWindow(object):            def __init__(self):                self.left = SortedList()                self.right = SortedList()                self.total1 = self.total2 = 0             def add(self, val):                if not self.left or val <= self.left[-1]:                    self.left.add(val)                    self.total1 += val                else:                    self.right.add(val)                    self.total2 += val                self.rebalance()             def remove(self, val):                if val <= self.left[-1]:                    self.left.remove(val)                    self.total1 -= val                else:                    self.right.remove(val)                    self.total2 -= val                self.rebalance()             def rebalance(self):                if len(self.left) < len(self.right):                    self.total2 -= self.right[0]                    self.total1 += self.right[0]                    self.left.add(self.right[0])                    self.right.pop(0)                elif len(self.left) > len(self.right)+1:                    self.total1 -= self.left[-1]                    self.total2 += self.left[-1]                    self.right.add(self.left[-1])                    self.left.pop()             def median(self):                return self.left[-1]          INF = float("inf")        sw = SlidingWindow()        cost = [INF]*(len(nums)+1)        for i in xrange(len(nums)):            if i-x >= 0:                sw.remove(nums[i-x])            sw.add(nums[i])            if i >= x-1:                cost[i+1] = (sw.median()*len(sw.left)-sw.total1) + (sw.total2-sw.median()*len(sw.right))        dp = [0]*(len(nums)+1)        for i in xrange(k):            new_dp = [INF]*(len(nums)+1)            for j in xrange((i+1)*x, len(nums)+1):                new_dp[j] = min(new_dp[j-1], dp[j-x]+cost[j])            dp = new_dp        return dp[-1]  # Time:  O(nlogx + k * n)# Space: O(n)import heapqimport collections  # two heaps, dpclass Solution2(object):    def minOperations(self, nums, x, k):        """        :type nums: List[int]        :type x: int        :type k: int        :rtype: int        """        class LazyHeap(object):            def __init__(self, sign):                self.heap = []                self.to_remove = collections.defaultdict(int)                self.cnt = 0                self.sign = sign             def push(self, val):                heapq.heappush(self.heap, self.sign*val)             def full_remove(self):                result = []                for x in self.heap:                    if x not in self.to_remove:                        result.append(x)                        continue                    self.to_remove[x] -= 1                    if not self.to_remove[x]:                        del self.to_remove[x]                self.heap[:] = result                heapq.heapify(self.heap)                def remove(self, val):                self.to_remove[self.sign*val] += 1                self.cnt += 1                if self.cnt > len(self.heap)-self.cnt:                    self.full_remove()                    self.cnt = 0             def pop(self):                self.remove(self.top())             def top(self):                while self.heap and self.heap[0] in self.to_remove:                    self.to_remove[self.heap[0]] -= 1                    self.cnt -= 1                    if self.to_remove[self.heap[0]] == 0:                        del self.to_remove[self.heap[0]]                    heapq.heappop(self.heap)                return self.sign*self.heap[0]             def __len__(self):                return len(self.heap)-self.cnt          class SlidingWindow(object):            def __init__(self):                self.left = LazyHeap(-1)   # max heap                self.right = LazyHeap(+1)  # min heap                self.total1 = self.total2 = 0             def add(self, val):                if not self.left or val <= self.left.top():                    self.left.push(val)                    self.total1 += val                else:                    self.right.push(val)                    self.total2 += val                self.rebalance()             def remove(self, val):                if val <= self.left.top():                    self.left.remove(val)                    self.total1 -= val                else:                    self.right.remove(val)                    self.total2 -= val                self.rebalance()             def rebalance(self):                if len(self.left) < len(self.right):                    self.total2 -= self.right.top()                    self.total1 += self.right.top()                    self.left.push(self.right.top())                    self.right.pop()                elif len(self.left) > len(self.right)+1:                    self.total1 -= self.left.top()                    self.total2 += self.left.top()                    self.right.push(self.left.top())                    self.left.pop()             def median(self):                return self.left.top()          INF = float("inf")        sw = SlidingWindow()        cost = [INF]*(len(nums)+1)        for i in xrange(len(nums)):            if i-x >= 0:                sw.remove(nums[i-x])            sw.add(nums[i])            if i >= x-1:                cost[i+1] = (sw.median()*len(sw.left)-sw.total1) + (sw.total2-sw.median()*len(sw.right))        dp = [0]*(len(nums)+1)        for i in xrange(k):            new_dp = [INF]*(len(nums)+1)            for j in xrange((i+1)*x, len(nums)+1):                new_dp[j] = min(new_dp[j-1], dp[j-x]+cost[j])            dp = new_dp        return dp[-1]

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