线段树

线段树(区间树)

• 使用线段树
  • 对于给定的区间
  • 更新区间中的一个元素或者一个区间的值
  • 查询一个区间[i,j]的最大值,最小值,或者区间数字和

线段树的实现

public class SegmentTree<E>{
    
    private E[] data;

    private E[] tree;

    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];

        buildSegmentTree(0,0,data.length-1);

    }

    // 在treeIndex的位置创建区间[l,r]的线段树
    private void buildSegmentTree(int treeIndex,int l,int r){
        if(l==r){
            tree[treeIndex] = data[l];
        }

        int leftTreeIndex = leftChild(treeIndex);
        int rightTreeIndex = rightChild(treeIndex);

        int mid = l+(r-l)/2;
        buildSegmentTree(leftTreeIndex,l,mid);
        buildSegmentTree(rightTreeIndex,mid+1,r);

        tree[treeIndex] = merger.merge(tree[leftTreeIndex],tree[rightTreeIndex]);
    }

    // 返回完全二叉树的数组表示中,一个索引所表示的元素的左孩子节点的索引
    private int leftChild(int index){
        return 2*index+1;
    }

    // 返回完全二叉树的数组表示中,一个索引所表示的元素的右孩子节点的索引
    private int rightChild(int index){
        return 2*index+2;
    }

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

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

    // 返回区间[queryL,queryR]的值
    // O(logn)
    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);
    }

    // 在以treeIndex为根的线段树中[l,r]的范围里,搜索区间[queryL,queryR]的值
    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 leftTreeIndex = leftChild(treeIndex);
        int rightTreeIndex = rightChild(treeIndex);

        if(queryL>=mid+1){
            return query(rightTreeIndex,mid+1,r,queryL,queryR);
        }else if(queryR<=mid){
            return query(leftTreeIndex,l,mid,queryL,queryR);
        }

        E leftResult = query(leftTreeIndex,l,mid,queryL,mid);
        E rightResult = query(rightTreeIndex,mid+1,r,mid+1,queryR);
        return merger.merge(leftResult,rightResult);
    }

    // 将index位置的值更新为e
    // O(logn)
    public void set(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);
    }

    // 在以treeIndex为根的线段树中更新index的值e
    private void set(int treeIndex,int l,int r,int index,E e){
        if(l==r){
            tree[treeIndex]=e;
            return;
        }

        int mid = l+(r-l)/2;
        int leftTreeIndex = leftChild(treeIndex);
        int rightTreeIndex = rightChild(treeIndex);
        if(index>=mid+1){
            set(rightTreeIndex,mid+1,r,index,e);
        }else{
            set(leftTreeIndex,l,mid,index,e);
        }

        tree[treeIndex] = merger.merge(leftTreeIndex,rightTreeIndex);

    }
}

public interface Merger<E>(){
    E merge(E a, E b);
}

线段树应用

• 区域和检索-不可变

扩展

• 对区间更新
• 二维线段树
• 动态线段树
• 树状数组