线段树(区间树)

偏执的太偏执、 2021-09-23 22:36 420阅读 0赞

# 为什么使用线段树？ #

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# 什么是线段树？ #

**每一个节点存储的是一个区间中相应统计值**   
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQzMjI5NTQz_size_16_color_FFFFFF_t_70 1]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQzMjI5NTQz_size_16_color_FFFFFF_t_70 2]

## 最坏情况“n=2的k次方+1” 举例如下：可以看到叶子节点存在很多null节点，而也就是说存在一定的浪费，但为了满子元素全部存放只能这样。 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQzMjI5NTQz_size_16_color_FFFFFF_t_70 3]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQzMjI5NTQz_size_16_color_FFFFFF_t_70 4]![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQzMjI5NTQz_size_16_color_FFFFFF_t_70 5]

## 查询操作 ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQzMjI5NTQz_size_16_color_FFFFFF_t_70 6]

# 全局代码实现 #

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];
            buildSegmentTree(0, 0, arr.length - 1);
        }
    
        // 在treeIndex的位置创建表示区间[l...r]的线段树
        private void buildSegmentTree(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) / 2;
            int mid = l + (r - l) / 2;
            buildSegmentTree(leftTreeIndex, l, mid);
            buildSegmentTree(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];
        }
    
        // 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;
        }
    
        // 返回区间[queryL, queryR]的值
        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;
            // treeIndex的节点分为[l...mid]和[mid+1...r]两部分
    
            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);
        }
    
        @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(", ");
            }
            res.append(']');
            return res.toString();
        }
    }

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