堆積與優先佇列

def solve_with_heap(nums, k=None):
    import heapq
    
    # Create heap (min heap by default in Python)
    heap = []
    
    # Build heap approach 1: Insert elements one by one
    for num in nums:
        heapq.heappush(heap, num)
    
    # Build heap approach 2: Heapify existing array
    # heapq.heapify(nums)  # O(N) time
    
    # Access min element (don't remove): heap[0]
    # Remove min element: heapq.heappop(heap)
    # Insert element: heapq.heappush(heap, value)
    
    # For max heap, use negative values
    # max_heap = [-x for x in nums]
    # heapq.heapify(max_heap)
    # max_val = -max_heap[0]  # Get max without removing
    # max_val = -heapq.heappop(max_heap)  # Remove and get max
    
    return heap

// Java Universal Template
public class HeapSolution {
    public void solveWithHeap(int[] nums, int k) {
        // Min Heap
        PriorityQueue<Integer> minHeap = new PriorityQueue<>();
        
        // Max Heap
        PriorityQueue<Integer> maxHeap = new PriorityQueue<>((a, b) -> Integer.compare(b, a));
        
        // Add elements
        for (int num : nums) {
            minHeap.offer(num);
        }
        
        // Access min: minHeap.peek()
        // Remove min: minHeap.poll()
        // Add element: minHeap.offer(value)
    }
}

def find_kth_largest(nums, k):
    import heapq

    # Method 1: Min heap of size k
    heap = []
    for num in nums:
        if len(heap) < k:
            heapq.heappush(heap, num)
        elif num > heap[0]:
            heapq.heapreplace(heap, num)

    return heap[0]  # kth largest

def find_kth_smallest(nums, k):
    import heapq

    # Method 1: Max heap of size k (use negative values)
    heap = []
    for num in nums:
        if len(heap) < k:
            heapq.heappush(heap, -num)
        elif num < -heap[0]:
            heapq.heapreplace(heap, -num)

    return -heap[0]  # kth smallest

# python
# LC 1985 - Find the Kth Largest Integer in the Array
# time = O(N log k), space = O(k)
# IDEA: kth largest -> min-heap of size k; key = (len, s) makes string order == numeric order
import heapq

def kthLargestNumber(nums, k):
    heap = []
    for s in nums:
        heapq.heappush(heap, (len(s), s))
        if len(heap) > k:
            heapq.heappop(heap)
    return heap[0][1]

// java
// LC 1985 - Find the Kth Largest Integer in the Array
// time = O(N log k), space = O(k)
// IDEA: min-heap of size k; comparator = length first, then lexicographic
public String kthLargestNumber(String[] nums, int k) {
    PriorityQueue<String> minHeap = new PriorityQueue<>(
        (a, b) -> a.length() != b.length() ? Integer.compare(a.length(), b.length()) : a.compareTo(b));

    for (String s : nums) {
        minHeap.offer(s);
        if (minHeap.size() > k) minHeap.poll();
    }
    return minHeap.peek();
}

def top_k_frequent(nums, k):
    from collections import Counter
    import heapq
    
    # Count frequencies
    count = Counter(nums)
    
    # Method 1: Min heap approach
    heap = []
    for num, freq in count.items():
        heapq.heappush(heap, (freq, num))
        if len(heap) > k:
            heapq.heappop(heap)
    
    return [item[1] for item in heap]
    
    # Method 2: Max heap approach
    # heap = [(-freq, num) for num, freq in count.items()]
    # heapq.heapify(heap)
    # return [heapq.heappop(heap)[1] for _ in range(k)]

def merge_k_sorted_arrays(arrays):
    import heapq
    
    heap = []
    result = []
    
    # Initialize heap with first element from each array
    for i, arr in enumerate(arrays):
        if arr:  # Check if array is not empty
            heapq.heappush(heap, (arr[0], i, 0))
    
    while heap:
        val, array_idx, element_idx = heapq.heappop(heap)
        result.append(val)
        
        # Add next element from same array
        if element_idx + 1 < len(arrays[array_idx]):
            next_val = arrays[array_idx][element_idx + 1]
            heapq.heappush(heap, (next_val, array_idx, element_idx + 1))
    
    return result

class MedianFinder:
    def __init__(self):
        import heapq
        self.small = []  # max heap (use negative values)
        self.large = []  # min heap
    
    def addNum(self, num):
        import heapq
        
        # Add to appropriate heap
        if len(self.small) == len(self.large):
            heapq.heappush(self.large, -heapq.heappushpop(self.small, -num))
        else:
            heapq.heappush(self.small, -heapq.heappushpop(self.large, num))
    
    def findMedian(self):
        if len(self.small) == len(self.large):
            return (self.large[0] - self.small[0]) / 2.0
        else:
            return float(self.large[0])

maxHeap = (-value, index)     minHeap = (value, index)
stale  <=>  index < left      (the element has fallen out of the window)

# python
# LC 1438 - Longest Continuous Subarray With Absolute Diff Less Than or Equal to Limit
# time = O(N log N), space = O(N)
# IDEA: two heaps hold window max/min; shrink past the older extreme; purge stale indices lazily
import heapq

class Solution(object):
    def longestSubarray(self, nums, limit):
        max_h = []   # max-heap: (-val, idx)
        min_h = []   # min-heap: ( val, idx)
        left = 0
        res = 0

        for i, v in enumerate(nums):
            heapq.heappush(max_h, (-v, i))
            heapq.heappush(min_h, (v, i))

            # window invalid -> must drop at least one of the two extremes
            while -max_h[0][0] - min_h[0][0] > limit:
                # NOTE !!! move left PAST the earlier of the two extreme indices
                left = min(max_h[0][1], min_h[0][1]) + 1
                # lazy delete: anything left of the window is stale
                while max_h[0][1] < left:
                    heapq.heappop(max_h)
                while min_h[0][1] < left:
                    heapq.heappop(min_h)

            res = max(res, i - left + 1)

        return res

// java
// LC 1438 - Longest Continuous Subarray With Absolute Diff Less Than or Equal to Limit
// time = O(N log N), space = O(N)
// IDEA: max-heap + min-heap of {val, idx}; shrink past older extreme; lazy-purge stale indices
public int longestSubarray(int[] nums, int limit) {
    PriorityQueue<int[]> maxH = new PriorityQueue<>((a, b) -> Integer.compare(b[0], a[0]));  // {val, idx}
    PriorityQueue<int[]> minH = new PriorityQueue<>((a, b) -> Integer.compare(a[0], b[0]));

    int left = 0, res = 0;

    for (int i = 0; i < nums.length; i++) {
        maxH.offer(new int[]{nums[i], i});
        minH.offer(new int[]{nums[i], i});

        while (maxH.peek()[0] - minH.peek()[0] > limit) {
            left = Math.min(maxH.peek()[1], minH.peek()[1]) + 1;
            while (maxH.peek()[1] < left) maxH.poll();
            while (minH.peek()[1] < left) minH.poll();
        }

        res = Math.max(res, i - left + 1);
    }
    return res;
}

# Python - Find K largest elements
def topKElements(nums, k):
    import heapq
    
    # Min heap of size k for k largest
    min_heap = []
    
    for num in nums:
        heapq.heappush(min_heap, num)
        if len(min_heap) > k:
            heapq.heappop(min_heap)
    
    return min_heap  # Contains k largest elements

# With custom key for frequency
def topKFrequent(nums, k):
    from collections import Counter
    import heapq
    
    count = Counter(nums)
    # Use negative count for max heap effect
    return heapq.nlargest(k, count.keys(), key=count.get)

// Java - Top K elements with frequency
public int[] topKFrequent(int[] nums, int k) {
    Map<Integer, Integer> map = new HashMap<>();
    for (int n : nums) {
        map.put(n, map.getOrDefault(n, 0) + 1);
    }
    
    // Min heap based on frequency
    PriorityQueue<Integer> pq = new PriorityQueue<>(
        (a, b) -> Integer.compare(map.get(a), map.get(b))
    );
    
    for (int key : map.keySet()) {
        pq.add(key);
        if (pq.size() > k) {
            pq.poll();
        }
    }
    
    int[] result = new int[k];
    for (int i = k - 1; i >= 0; i--) {
        result[i] = pq.poll();
    }
    return result;
}

# Python - Merge K sorted lists
def mergeKSortedLists(lists):
    import heapq
    
    min_heap = []
    
    # Initialize with first element from each list
    for i, lst in enumerate(lists):
        if lst:
            heapq.heappush(min_heap, (lst[0], i, 0))
    
    result = []
    while min_heap:
        val, list_idx, elem_idx = heapq.heappop(min_heap)
        result.append(val)
        
        # Add next element from same list
        if elem_idx + 1 < len(lists[list_idx]):
            next_val = lists[list_idx][elem_idx + 1]
            heapq.heappush(min_heap, (next_val, list_idx, elem_idx + 1))
    
    return result

// Java - Merge K sorted arrays
public int[] mergeKSortedArrays(int[][] arrays) {
    PriorityQueue<int[]> pq = new PriorityQueue<>(
        (a, b) -> Integer.compare(a[0], b[0])  // Compare values
    );
    
    int totalSize = 0;
    // Initialize PQ with first element from each array
    for (int i = 0; i < arrays.length; i++) {
        if (arrays[i].length > 0) {
            pq.offer(new int[]{arrays[i][0], i, 0});
            totalSize += arrays[i].length;
        }
    }
    
    int[] result = new int[totalSize];
    int idx = 0;
    
    while (!pq.isEmpty()) {
        int[] curr = pq.poll();
        result[idx++] = curr[0];
        
        int arrIdx = curr[1];
        int elemIdx = curr[2];
        
        if (elemIdx + 1 < arrays[arrIdx].length) {
            pq.offer(new int[]{
                arrays[arrIdx][elemIdx + 1], 
                arrIdx, 
                elemIdx + 1
            });
        }
    }
    
    return result;
}

// Java - Two heaps for median
class MedianFinder {
    private PriorityQueue<Integer> small;  // Max heap
    private PriorityQueue<Integer> large;  // Min heap
    
    public MedianFinder() {
        small = new PriorityQueue<>(Collections.reverseOrder());
        large = new PriorityQueue<>();
    }
    
    public void addNum(int num) {
        small.offer(num);
        
        // Balance property
        if (!small.isEmpty() && !large.isEmpty() && 
            small.peek() > large.peek()) {
            large.offer(small.poll());
        }
        
        // Size property
        if (small.size() > large.size() + 1) {
            large.offer(small.poll());
        }
        if (large.size() > small.size() + 1) {
            small.offer(large.poll());
        }
    }
    
    public double findMedian() {
        if (small.size() > large.size()) {
            return small.peek();
        }
        if (large.size() > small.size()) {
            return large.peek();
        }
        return (small.peek() + large.peek()) / 2.0;
    }
}

# Python - Meeting rooms (minimum rooms needed)
def minMeetingRooms(intervals):
    import heapq
    
    if not intervals:
        return 0
    
    # Sort by start time
    intervals.sort(key=lambda x: x[0])
    
    # Min heap to track end times
    heap = []
    heapq.heappush(heap, intervals[0][1])
    
    for i in range(1, len(intervals)):
        # If current meeting starts after earliest end
        if intervals[i][0] >= heap[0]:
            heapq.heappop(heap)
        
        # Add current meeting end time
        heapq.heappush(heap, intervals[i][1])
    
    return len(heap)

// Java - Interval scheduling
public int minMeetingRooms(int[][] intervals) {
    if (intervals.length == 0) return 0;
    
    // Sort by start time
    Arrays.sort(intervals, (a, b) -> Integer.compare(a[0], b[0]));
    
    // Min heap for end times
    PriorityQueue<Integer> pq = new PriorityQueue<>();
    pq.offer(intervals[0][1]);
    
    for (int i = 1; i < intervals.length; i++) {
        // Room becomes free
        if (intervals[i][0] >= pq.peek()) {
            pq.poll();
        }
        pq.offer(intervals[i][1]);
    }
    
    return pq.size();
}

// java
// LC 1353 - Maximum Number of Events That Can Be Attended
// IDEA: sort by start day; each day push all events that opened, drop expired ones,
//       then attend the one with the earliest end day (min-heap)
// time = O(n log n), space = O(n)
public int maxEvents(int[][] events) {
    Arrays.sort(events, (a, b) -> Integer.compare(a[0], b[0]));
    PriorityQueue<Integer> pq = new PriorityQueue<>();   // end days of open events
    int i = 0, n = events.length, res = 0, day = 0;

    while (i < n || !pq.isEmpty()) {
        // idle -> jump to the next start day, otherwise advance one day
        day = pq.isEmpty() ? events[i][0] : day + 1;

        while (i < n && events[i][0] <= day) pq.offer(events[i++][1]);   // now open
        while (!pq.isEmpty() && pq.peek() < day) pq.poll();              // expired

        if (!pq.isEmpty()) { pq.poll(); res++; }   // attend earliest-ending event
    }
    return res;
}

# Python - Custom priority with multiple criteria
class Task:
    def __init__(self, name, priority, deadline):
        self.name = name
        self.priority = priority
        self.deadline = deadline
    
    def __lt__(self, other):
        # Higher priority first, then earlier deadline
        if self.priority != other.priority:
            return self.priority > other.priority
        return self.deadline < other.deadline

def processTasks(tasks):
    import heapq
    
    heap = []
    for task in tasks:
        heapq.heappush(heap, task)
    
    result = []
    while heap:
        task = heapq.heappop(heap)
        result.append(task.name)
    
    return result

// Java - Custom comparator for complex ordering
class Task {
    String name;
    int priority;
    int deadline;
    
    Task(String name, int priority, int deadline) {
        this.name = name;
        this.priority = priority;
        this.deadline = deadline;
    }
}

public List<String> processTasks(List<Task> tasks) {
    PriorityQueue<Task> pq = new PriorityQueue<>((a, b) -> {
        // Higher priority first
        if (a.priority != b.priority) {
            return Integer.compare(b.priority, a.priority);
        }
        // Earlier deadline first
        return Integer.compare(a.deadline, b.deadline);
    });
    
    for (Task task : tasks) {
        pq.offer(task);
    }
    
    List<String> result = new ArrayList<>();
    while (!pq.isEmpty()) {
        result.add(pq.poll().name);
    }
    
    return result;
}

// Java - Longest Happy String (LC 1405) / Reorganize String (LC 767)
// IDEA: Max-heap by count; two-case loop:
//   Case 1: top char would create 3 consecutive → use 2nd, put 1st back
//   Case 2: safe → use top char directly
// time = O((a+b+c) * log(3)) = O(n), space = O(1) heap size bounded by alphabet

class ValCnt {
    char val;
    int cnt;
    ValCnt(char val, int cnt) { this.val = val; this.cnt = cnt; }
}

public String longestDiverseString(int a, int b, int c) {
    PriorityQueue<ValCnt> pq = new PriorityQueue<>((x, y) -> Integer.compare(y.cnt, x.cnt));
    if (a > 0) pq.add(new ValCnt('a', a));
    if (b > 0) pq.add(new ValCnt('b', b));
    if (c > 0) pq.add(new ValCnt('c', c));

    StringBuilder sb = new StringBuilder();

    while (!pq.isEmpty()) {
        ValCnt first = pq.poll();
        int len = sb.length();

        // Case 1: adding `first` would create 3 consecutive → use second instead
        if (len >= 2
                && sb.charAt(len - 1) == first.val
                && sb.charAt(len - 2) == first.val) {

            if (pq.isEmpty()) break;          // no alternative → stop

            ValCnt second = pq.poll();        // use 2nd most frequent
            sb.append(second.val);
            second.cnt--;

            if (second.cnt > 0) pq.add(second);
            pq.add(first);                    // first was NOT used, put it back

        // Case 2: safe to use the most frequent character
        } else {
            sb.append(first.val);
            first.cnt--;
            if (first.cnt > 0) pq.add(first);
        }
    }

    return sb.toString();
}

// Only Case 1 check changes: len >= 1 && sb.charAt(len-1) == first.val
// Everything else is identical to the template above.

// Java - Rearrange String k Distance Apart (LC 358)
// IDEA: Max-heap picks most frequent available char;
//       cooldown queue locks used chars for k steps.
//
// Flow: PQ → poll → append → cooldown.offer([char, releaseTime])
//       when time == releaseTime → move back to PQ
//
// time = O(n log 26) = O(n), space = O(26) = O(1)

public String rearrangeString(String s, int k) {
    if (k <= 1) return s;

    int[] freq = new int[26];
    for (char c : s.toCharArray()) {
        freq[c - 'a']++;
    }

    // Max-heap ordered by remaining frequency
    PriorityQueue<Integer> pq = new PriorityQueue<>((a, b) -> Integer.compare(freq[b], freq[a]));
    for (int i = 0; i < 26; i++) {
        if (freq[i] > 0) pq.offer(i);
    }

    // Cooldown queue: [charIndex, remainingCount]
    // Size reaches k → front element has cooled for k steps → re-enable
    Queue<int[]> cooldown = new LinkedList<>();
    StringBuilder res = new StringBuilder();

    while (!pq.isEmpty()) {
        int idx = pq.poll();
        res.append((char) ('a' + idx));
        freq[idx]--;

        // Enter cooldown with current remaining count
        cooldown.offer(new int[]{idx, freq[idx]});

        // Release from cooldown after k steps
        if (cooldown.size() == k) {
            int[] ready = cooldown.poll();
            if (ready[1] > 0) {
                pq.offer(ready[0]);  // re-add to heap
            }
        }
    }

    return res.length() == s.length() ? res.toString() : "";
}

# python
# the size-k idiom, written once
import heapq

def k_largest(nums, k):                 # time = O(N log k), space = O(k)
    h = []
    for v in nums:
        if len(h) < k:
            heapq.heappush(h, v)
        elif v > h[0]:                  # peek, then replace in one sift
            heapq.heapreplace(h, v)
    return h                            # h[0] == the kth largest

def k_smallest(nums, k):                # max-heap = negate on the way in and out
    h = []
    for v in nums:
        heapq.heappush(h, -v)
        if len(h) > k:
            heapq.heappop(h)
    return [-x for x in h]              # -h[0] == the kth smallest

import heapq

# Create max heap by negating values
max_heap = [-x for x in nums]
heapq.heapify(max_heap)

# Insert into max heap
heapq.heappush(max_heap, -val)

# Get max value (remember to negate back)
max_val = -max_heap[0]  # peek
max_val = -heapq.heappop(max_heap)  # pop

# Method 1: Using tuples (automatic comparison)
heap = []
heapq.heappush(heap, (priority, data))

# Method 2: Using custom class with __lt__
class Task:
    def __init__(self, priority, data):
        self.priority = priority
        self.data = data

    def __lt__(self, other):
        return self.priority < other.priority

heap = []
heapq.heappush(heap, Task(1, "high priority"))

# python
# a heap of (key, payload) compares the payload when keys tie -> unorderable types crash.
# push a monotone counter as the second element so the comparison never reaches the payload.
import heapq, itertools

counter = itertools.count()
heapq.heappush(pq, (priority, next(counter), payload))   # FIFO among equal priorities