数据结构和算法-06线段树-01

线段树

什么是线段树

线段树是一种** $二叉搜索树$ **，与 $\*\*区间树$ **相似，它将一个区间划分成一些单元区间，每个单元区间对应线段树的一个结点

$Segment Tree$ is a data structure that stores data about range of elements in nodes as a tree. It is mostly used to handle range queries with updates in an efficient manner.

线段树特性

线段树具有平衡二叉树的特性，节点有范围(可以用来记录范围最值，求和，平均值等)。

A segment tree is a binary tree with a leaf node for each element in the array. ---- 平衡二叉树
Each internal node represents a segment or a range of elements in the array. --- 节点表示范围
The root node represents ++the entire array++ . --- 根节点表示整棵树的范围
Each node stores information about the segment it represents, such as the sum or minimum of the elements in the segment. -- -每个节点存储线段的最大值，最小值

编号	线段树特性
1	平衡二叉树
2	节点表示范围
3	根节点表示整棵树的范围
4	每个节点存储线段的最大、最小值

线段树的实现-节点

创建节点SegmentNode

java 复制代码

**
 * 线段树节点
 */
public class SegmentNode {

    public SegmentNode left;
    public SegmentNode right;
    public int start;
    public int end;
    public int sum;

    public SegmentNode(int start, int end) {
        this.start = start;
        this.end = end;
    }

    @Override
    public String toString() {
        return "SegmentNode{" +
                "left=" + left +
                ", right=" + right +
                ", start=" + start +
                ", end=" + end +
                ", sum=" + sum +
                '}';
    }
}

构建线段树

java 复制代码

public SegmentNode buildTree(int start, int end) {
    if (start > end) {
        return null;
    }
    SegmentNode newNode = new SegmentNode(start, end);

    if (start == end) {
        newNode.sum = elements[start];
        return newNode;
    }

    int mid = start + (end - start) / 2;
    newNode.left = buildTree(start, mid);
    newNode.right = buildTree(mid + 1, end);
    return newNode;
}

查询指定线段树数据

java 复制代码

public long query(SegmentNode root, int queryStart, int queryEnd){
    if(queryStart > root.end || queryEnd < root.start){
        return  0;
    }
    if(queryStart <= root.start && queryEnd >= root.end){
        return root.sum;
    }

    long leftSum = query(root.left, queryStart, queryEnd);
    long rightSum = query(root.right, queryStart,queryEnd);
    return leftSum +rightSum;
}

更新线段树

在线段树中，pushDown() 方法通常用于实现懒更新 （Lazy Propagation）策略。懒更新策略允许我们在++更新操作时避免不必要的子树遍历，只在真正需要时才将更新值下推到子节点++。在节点中添加lazy属性记录lazy值。

如果我们每进行一次加的操作，就将全部线段树更改一边，时间复杂度会很高。因此，我们需要进行一个延迟加和的操作。

$思路$ ：如果 $left,right$ 区间增加 a,在查询时,就可以把 $left,right$ 区间标记的增加量推下去就可以直接求值了。

这时候，我们需要记录一个懒标记lazy，来记录这个区间增加量。

pushDown()

创建pushDown( )方法，如果子节点不为空，更新子节点的lazy值，即记录子节点的lazy值。

java 复制代码

private void pushDown(SegmentNode node) {
    if (node.left == null || node.right == null) return;

    if (node.lazy != 0) {
        int mid = node.start + (node.end - node.start)/2;
        node.left.sum += node.lazy * (mid - node.start + 1);
        node.left.lazy += node.lazy;

        node.right.sum += node.lazy * (node.end -mid);
        node.right.lazy += node.lazy;
        node.lazy = 0;
    }

}

更新节点

java 复制代码

public void update(SegmentNode currNode,
                          int updateStart,
                          int updateEnd,
                          int updateVal) {
    //不在范围内，更新失败
    if (updateStart > currNode.end || updateEnd < currNode.start) return ;
    //完全闭合区域
    if (updateStart >= currNode.start && updateEnd <= currNode.end) {
        currNode.sum += (currNode.end - currNode.start + 1) * updateVal;
        currNode.lazy += updateVal;
        return ;
    }

    pushDown(currNode);
    
    update(currNode.left, updateStart, updateEnd, updateVal);
    update(currNode.right, updateStart, updateEnd, updateVal);

    currNode.sum = currNode.left.sum + currNode.right.sum;
    return ;
}

打印线段树

java 复制代码

public void printTree(SegmentTreeNode root, String prefix) {
    System.out.println(
        prefix + "Node [" + root.start + ", " + root.end + "]: value = " + root.sum + ", lazy = " + root.lazy);

    if (root.left != null) {
        printTree(root.left, prefix + "  "); // 增加前缀空格以便形成缩进
    }
    if (root.right != null) {
        printTree(root.right, prefix + "  ");
    }
}

完整代码实现

java 复制代码

public class SegmentNode {

    public SegmentNode left;
    public SegmentNode right;
    public int start;
    public int end;
    public int sum;
    public int lazy;

    public SegmentNode(int start, int end) {
        this.start = start;
        this.end = end;
    }

    @Override
    public String toString() {
        return "SegmentNode{" +
                ", start=" + start +
                ", end=" + end +
                ", sum=" + sum +
                '}';
    }
}

java 复制代码

package com.training.segment;

public class SegmentTree {
    private int[] elements;

    public SegmentTree(int[] elements) {
        this.elements = elements;
    }
    public SegmentNode buildTree(int start, int end) {
        if (start > end) {
            return null;
        }
        SegmentNode newNode = new SegmentNode(start, end);

        if (start == end) {
            newNode.sum = elements[start];
            return newNode;
        }

        int mid = start + (end - start) / 2;
        newNode.left = buildTree(start, mid);
        newNode.right = buildTree(mid + 1, end);
        newNode.sum = (newNode.left == null ? 0 : newNode.left.sum) + (newNode.right == null ? 0 : newNode.right.sum);

        return newNode;
    }

    public long query(SegmentNode root, int queryStart, int queryEnd) {
        if (queryStart > root.end || queryEnd < root.start) {
            return 0;
        }
        if (queryStart <= root.start && queryEnd >= root.end) {
            return root.sum;
        }

        long leftSum = query(root.left, queryStart, queryEnd);
        long rightSum = query(root.right, queryStart, queryEnd);
        return leftSum + rightSum;
    }

    private void pushDown(SegmentNode node) {
        if (node.left == null || node.right == null) return;

        if (node.lazy != 0) {
            int mid = node.start + (node.end - node.start) / 2;
            node.left.sum += node.lazy * (mid - node.start + 1);
            node.left.lazy += node.lazy;

            node.right.sum += node.lazy * (node.end - mid);
            node.right.lazy += node.lazy;

            node.lazy = 0;
        }

    }

    public void update(SegmentNode currNode,
                       int updateStart,
                       int updateEnd,
                       int updateVal) {
        //不在范围内，更新失败
        if (updateStart > currNode.end || updateEnd < currNode.start) return;
        //完全闭合区域
        if (updateStart <= currNode.start && updateEnd >= currNode.end) {
            currNode.sum += (currNode.end - currNode.start + 1) * updateVal;
            currNode.lazy += updateVal;
            return;
        }

        pushDown(currNode);

        update(currNode.left, updateStart, updateEnd, updateVal);
        update(currNode.right, updateStart, updateEnd, updateVal);
        int leftSum = currNode.left == null ? 0 : currNode.left.sum;
        int rightSum = currNode.right == null ? 0 : currNode.right.sum;
        currNode.sum = leftSum + rightSum;
    }

    public void printTree(SegmentNode root, String prefix) {
        System.out.println(
                prefix + "Node [" + root.start + ", " + root.end + "]: value = "
                        + root.sum);

        if (root.left != null) {
            printTree(root.left, prefix + "  "); // 增加前缀空格以便形成缩进
        }
        if (root.right != null) {
            printTree(root.right, prefix + "  ");
        }
    }

    public static void main(String[] args) {
        int[] data = {1, 3, 5, 7, 9, 11};
        SegmentTree tree = new SegmentTree(data);
        SegmentNode root = tree.buildTree(0, data.length - 1);
        tree.printTree(root, "");

        tree.update(root, 0, 2, 3);
        System.out.println("-------------------------------------------------------");
        tree.printTree(root, "");
        System.out.println(tree.query(root, 2, 2));
    }
}

测试结果

java 复制代码

Node [0, 5]: value = 36
  Node [0, 2]: value = 9
    Node [0, 1]: value = 4
      Node [0, 0]: value = 1
      Node [1, 1]: value = 3
    Node [2, 2]: value = 5
  Node [3, 5]: value = 27
    Node [3, 4]: value = 16
      Node [3, 3]: value = 7
      Node [4, 4]: value = 9
    Node [5, 5]: value = 11
-------------------------------------------------------
Node [0, 5]: value = 45
  Node [0, 2]: value = 18
    Node [0, 1]: value = 4
      Node [0, 0]: value = 1
      Node [1, 1]: value = 3
    Node [2, 2]: value = 5
  Node [3, 5]: value = 27
    Node [3, 4]: value = 16
      Node [3, 3]: value = 7
      Node [4, 4]: value = 9
    Node [5, 5]: value = 11
5