線段樹

定義

線段樹是一種二叉搜索樹，與區間樹相似，它將一個區間劃分成一些單元區間，每個單元區間對應線段樹中的一個葉結點。

使用線段樹可以快速的查找某一個節點在若干條線段中出現的次數，時間複雜度爲O(logN)。而未優化的空間複雜度爲2N，實際應用時一般還要開4N的數組以免越界，因此有時需要離散化讓空間壓縮。

代碼實現

package com.company;

public class SegmentTree<E> {

    private E[] tree;
    private E[] data;
    private Merger<E> merger;

    public SegmentTree(E[] arr, Merger<E> merger) {
        this.merger = merger;
        data = (E[]) new Object[arr.length];
        for (int i = 0; i < arr.length; i++) {
            data[i] = arr[i];
        }
        tree = (E[]) new Object[4 * arr.length];
        bulidSegmentTree(0, 0, data.length - 1);
    }

    // 在treeIndex的位置創建表示區間[l...r]的線段樹
    private void bulidSegmentTree(int treeIndex, int l, int r) {
        if (l == r) {
            tree[treeIndex] = data[l];
            return;
        }
        int leftTreeIndex = leftChild(treeIndex);
        int rightTreeIndex = rightChild(treeIndex);
        int mid = l + (r - l) / 2;
        bulidSegmentTree(leftTreeIndex, l, mid);
        bulidSegmentTree(rightTreeIndex, mid + 1, r);
        tree[treeIndex] = merger.merge(tree[leftTreeIndex], tree[rightTreeIndex]);
    }

    public int getSize() {
        return data.length;
    }

    public E get(int index) {
        if (index < 0 || index > data.length) {
            throw new IllegalArgumentException("Index is illegal.");
        }
        return data[index];
    }

    public int leftChild(int index) {
        return 2 * index + 1;
    }

    public int rightChild(int index) {
        return 2 * index + 2;
    }

    // 在tree中查找區間[l...r]
    public E query(int queryL, int queryR) {
        if (queryL < 0 || queryL >= data.length || queryR < 0 || queryR >= data.length || queryL > queryR) {
            throw new IllegalArgumentException("Index is illegal.");
        }
        return query(0, 0, data.length - 1, queryL, queryR);

    }

    private E query(int treeIndex, int l, int r, int queryL, int queryR) {
        if (l == queryL && r == queryR) {
            return tree[treeIndex];
        }
        int mid = l + (r - l) / 2;
        int leftChildIndex = leftChild(treeIndex);
        int rightChildIndex = rightChild(treeIndex);
        if (queryL >= mid + 1) {
            return query(rightChildIndex, mid + 1, r, queryL, queryR);
        } else if (queryR <= mid + 1) {
            return query(leftChildIndex, l, mid, queryL, queryR);
        } else {
            E leftResult = query(leftChildIndex, l, mid, queryL, mid);
            E rightResult = query(rightChildIndex, mid + 1, r, mid + 1, queryR);
            return merger.merge(leftResult, rightResult);
        }
    }

    public void set(int index, E e) {
        if (index < 0 || index >= data.length) {
            throw new IllegalArgumentException("Index is illegal.");
        }
        data[index] = e;
        set(0, 0, data.length - 1, index, e);
    }

    private void set(int treeIndex, int l, int r, int index, E e) {
        if (l == r) {
            data[treeIndex] = e;
            return;
        }
        int mid = l + (r - l) / 2;
        int leftChildIndex = leftChild(treeIndex);
        int rightChildIndex = rightChild(treeIndex);
        if (index >= mid + 1) {
            set(rightChildIndex, mid+1, r, index, e);
        } else {
            set(leftChildIndex, l, mid, index, e);
        }
        tree[treeIndex] = merger.merge(tree[leftChildIndex], tree[rightChildIndex]);
    }

    @Override
    public String toString() {
        StringBuilder res = new StringBuilder();
        res.append('[');
        for (int i = 0; i < tree.length; i++) {
            if (tree[i] != null) {
                res.append(tree[i]);
            } else {
                res.append("null");
            }
            if (i != tree.length - 1) {
                res.append(',');
            } else {
                res.append(']');
            }
        }
        return res.toString();
    }
}

區域和檢索 - 數組不可變

public class NumArray {

    private int[] sum;
    private int[] data;
    public NumArray(int[] nums) {
        data = new int[nums.length];
        for (int i = 0; i < data.length; i++) {
            data[i] = nums[i];
        }
        sum = new int[nums.length + 1];
        sum[0] = 0;
        for (int i = 1; i < sum.length; i++){
            sum[i] = sum[i - 1] + nums[i - 1];
        }
    }

    public void update(int index, int val) {
        data[index] = val;
        for (int i = index; i < sum.length ; i++) {
            sum[i] = sum[i - 1] + data[i - 1];
        }
    }

    public int sumRange(int i ,int j) {
        return sum[j + 1] - sum[i];
    }
}

區域和檢索 - 數組可修改

class NumArray {

    private int[] sum;
    private int[] data;

    public NumArray(int[] nums) {
        data = new int[nums.length];
        for (int i = 0; i < data.length; i++) {
            data[i] = nums[i];
        }
        sum = new int[nums.length + 1];
        sum[0] = 0;
        for (int i = 1; i < sum.length; i++){
            sum[i] = sum[i - 1] + nums[i - 1];
        }
    }

    public void update(int i, int val) {
        data[i] = val;
        for (int j = i + 1; j < sum.length ; j++) {
            sum[j] = sum[j - 1] + data[j - 1];
        }
    }

    public int sumRange(int i ,int j) {
        return sum[j + 1] - sum[i];
    }
}