给定一个字符串，构建它 后缀数组 我们已经讨论了以下两种构建后缀数组的方法：
null
请通读这些以获得基本的理解。 这里我们将看到如何使用后缀树在线性时间内构建后缀数组。 作为先决条件，我们必须知道如何以一种或另一种方式构建后缀树。 在这里，我们将使用Ukkonen的算法构建后缀树，如下所述： Ukkonen后缀树构造——第1部分 Ukkonen后缀树构造——第2部分 Ukkonen后缀树构造——第3部分 Ukkonen后缀树构造——第4部分 Ukkonen后缀树构造——第5部分 Ukkonen后缀树构造——第6部分 让我们考虑字符串ABCABXABCD。 它的后缀数组是：
0 6 3 1 7 4 2 8 9 5
让我们看看下图：
如果我们进行DFS遍历，按字典顺序访问边（我们在其他后缀树应用程序文章中也进行了同样的遍历），并在叶子上打印后缀索引，我们将得到以下结果：
10 0 6 3 1 7 4 2 8 9 5
“$”在词典编纂上小于[a-zA-Z]。 后缀索引10对应于带有“$”标签的边。 除了这个1 圣 后缀索引，所有其他数字的序列给出字符串的后缀数组。 所以，如果我们有一个字符串的后缀树，那么为了得到它的后缀数组，我们只需要按字典顺序进行DFS遍历，并将所有后缀索引存储在结果后缀数组中，除了1之外 圣 后缀索引。
C
// A C program to implement Ukkonen's Suffix Tree Construction // and then create suffix array in linear time #include <stdio.h> #include <string.h> #include <stdlib.h> #define MAX_CHAR 256 struct SuffixTreeNode { struct SuffixTreeNode *children[MAX_CHAR]; //pointer to other node via suffix link struct SuffixTreeNode *suffixLink; /*(start, end) interval specifies the edge, by which the node is connected to its parent node. Each edge will connect two nodes, one parent and one child, and (start, end) interval of a given edge will be stored in the child node. Lets say there are two nods A and B connected by an edge with indices (5, 8) then this indices (5, 8) will be stored in node B. */ int start; int *end; /*for leaf nodes, it stores the index of suffix for the path from root to leaf*/ int suffixIndex; }; typedef struct SuffixTreeNode Node; char text[100]; //Input string Node *root = NULL; //Pointer to root node /*lastNewNode will point to newly created internal node, waiting for it's suffix link to be set, which might get a new suffix link (other than root) in next extension of same phase. lastNewNode will be set to NULL when last newly created internal node (if there is any) got it's suffix link reset to new internal node created in next extension of same phase. */ Node *lastNewNode = NULL; Node *activeNode = NULL; /*activeEdge is represented as input string character index (not the character itself)*/ int activeEdge = -1; int activeLength = 0; // remainingSuffixCount tells how many suffixes yet to // be added in tree int remainingSuffixCount = 0; int leafEnd = -1; int *rootEnd = NULL; int *splitEnd = NULL; int size = -1; //Length of input string Node *newNode( int start, int *end) { Node *node =(Node*) malloc ( sizeof (Node)); int i; for (i = 0; i < MAX_CHAR; i++) node->children[i] = NULL; /*For root node, suffixLink will be set to NULL For internal nodes, suffixLink will be set to root by default in current extension and may change in next extension*/ node->suffixLink = root; node->start = start; node->end = end; /*suffixIndex will be set to -1 by default and actual suffix index will be set later for leaves at the end of all phases*/ node->suffixIndex = -1; return node; } int edgeLength(Node *n) { if (n == root) return 0; return *(n->end) - (n->start) + 1; } int walkDown(Node *currNode) { /*activePoint change for walk down (APCFWD) using Skip/Count Trick (Trick 1). If activeLength is greater than current edge length, set next internal node as activeNode and adjust activeEdge and activeLength accordingly to represent same activePoint*/ if (activeLength >= edgeLength(currNode)) { activeEdge += edgeLength(currNode); activeLength -= edgeLength(currNode); activeNode = currNode; return 1; } return 0; } void extendSuffixTree( int pos) { /*Extension Rule 1, this takes care of extending all leaves created so far in tree*/ leafEnd = pos; /*Increment remainingSuffixCount indicating that a new suffix added to the list of suffixes yet to be added in tree*/ remainingSuffixCount++; /*set lastNewNode to NULL while starting a new phase, indicating there is no internal node waiting for it's suffix link reset in current phase*/ lastNewNode = NULL; //Add all suffixes (yet to be added) one by one in tree while (remainingSuffixCount > 0) { if (activeLength == 0) activeEdge = pos; //APCFALZ // There is no outgoing edge starting with // activeEdge from activeNode if (activeNode->children] == NULL) { //Extension Rule 2 (A new leaf edge gets created) activeNode->children] = newNode(pos, &leafEnd); /*A new leaf edge is created in above line starting from an existing node (the current activeNode), and if there is any internal node waiting for it's suffix link get reset, point the suffix link from that last internal node to current activeNode. Then set lastNewNode to NULL indicating no more node waiting for suffix link reset.*/ if (lastNewNode != NULL) { lastNewNode->suffixLink = activeNode; lastNewNode = NULL; } } // There is an outgoing edge starting with activeEdge // from activeNode else { // Get the next node at the end of edge starting // with activeEdge Node *next = activeNode->children]; if (walkDown(next)) //Do walkdown { //Start from next node (the new activeNode) continue ; } /*Extension Rule 3 (current character being processed is already on the edge)*/ if (text[next->start + activeLength] == text[pos]) { //If a newly created node waiting for it's //suffix link to be set, then set suffix link //of that waiting node to current active node if (lastNewNode != NULL && activeNode != root) { lastNewNode->suffixLink = activeNode; lastNewNode = NULL; } //APCFER3 activeLength++; /*STOP all further processing in this phase and move on to next phase*/ break ; } /*We will be here when activePoint is in middle of the edge being traversed and current character being processed is not on the edge (we fall off the tree). In this case, we add a new internal node and a new leaf edge going out of that new node. This is Extension Rule 2, where a new leaf edge and a new internal node get created*/ splitEnd = ( int *) malloc ( sizeof ( int )); *splitEnd = next->start + activeLength - 1; //New internal node Node *split = newNode(next->start, splitEnd); activeNode->children] = split; //New leaf coming out of new internal node split->children] = newNode(pos, &leafEnd); next->start += activeLength; split->children] = next; /*We got a new internal node here. If there is any internal node created in last extensions of same phase which is still waiting for it's suffix link reset, do it now.*/ if (lastNewNode != NULL) { /*suffixLink of lastNewNode points to current newly created internal node*/ lastNewNode->suffixLink = split; } /*Make the current newly created internal node waiting for it's suffix link reset (which is pointing to root at present). If we come across any other internal node (existing or newly created) in next extension of same phase, when a new leaf edge gets added (i.e. when Extension Rule 2 applies is any of the next extension of same phase) at that point, suffixLink of this node will point to that internal node.*/ lastNewNode = split; } /* One suffix got added in tree, decrement the count of suffixes yet to be added.*/ remainingSuffixCount--; if (activeNode == root && activeLength > 0) //APCFER2C1 { activeLength--; activeEdge = pos - remainingSuffixCount + 1; } else if (activeNode != root) //APCFER2C2 { activeNode = activeNode->suffixLink; } } } void print( int i, int j) { int k; for (k=i; k<=j; k++) printf ( "%c" , text[k]); } //Print the suffix tree as well along with setting suffix index //So tree will be printed in DFS manner //Each edge along with it's suffix index will be printed void setSuffixIndexByDFS(Node *n, int labelHeight) { if (n == NULL) return ; if (n->start != -1) //A non-root node { //Print the label on edge from parent to current node //Uncomment below line to print suffix tree // print(n->start, *(n->end)); } int leaf = 1; int i; for (i = 0; i < MAX_CHAR; i++) { if (n->children[i] != NULL) { //Uncomment below two lines to print suffix index // if (leaf == 1 && n->start != -1) // printf(" [%d]", n->suffixIndex); //Current node is not a leaf as it has outgoing //edges from it. leaf = 0; setSuffixIndexByDFS(n->children[i], labelHeight + edgeLength(n->children[i])); } } if (leaf == 1) { n->suffixIndex = size - labelHeight; //Uncomment below line to print suffix index //printf(" [%d]", n->suffixIndex); } } void freeSuffixTreeByPostOrder(Node *n) { if (n == NULL) return ; int i; for (i = 0; i < MAX_CHAR; i++) { if (n->children[i] != NULL) { freeSuffixTreeByPostOrder(n->children[i]); } } if (n->suffixIndex == -1) free (n->end); free (n); } /*Build the suffix tree and print the edge labels along with suffixIndex. suffixIndex for leaf edges will be >= 0 and for non-leaf edges will be -1*/ void buildSuffixTree() { size = strlen (text); int i; rootEnd = ( int *) malloc ( sizeof ( int )); *rootEnd = - 1; /*Root is a special node with start and end indices as -1, as it has no parent from where an edge comes to root*/ root = newNode(-1, rootEnd); activeNode = root; //First activeNode will be root for (i=0; i<size; i++) extendSuffixTree(i); int labelHeight = 0; setSuffixIndexByDFS(root, labelHeight); } void doTraversal(Node *n, int suffixArray[], int *idx) { if (n == NULL) { return ; } int i=0; if (n->suffixIndex == -1) //If it is internal node { for (i = 0; i < MAX_CHAR; i++) { if (n->children[i] != NULL) { doTraversal(n->children[i], suffixArray, idx); } } } //If it is Leaf node other than "$" label else if (n->suffixIndex > -1 && n->suffixIndex < size) { suffixArray[(*idx)++] = n->suffixIndex; } } void buildSuffixArray( int suffixArray[]) { int i = 0; for (i=0; i< size; i++) suffixArray[i] = -1; int idx = 0; doTraversal(root, suffixArray, &idx); printf ( "Suffix Array for String " ); for (i=0; i<size; i++) printf ( "%c" , text[i]); printf ( " is: " ); for (i=0; i<size; i++) printf ( "%d " , suffixArray[i]); printf ( "" ); } // driver program to test above functions int main( int argc, char *argv[]) { strcpy (text, "banana$" ); buildSuffixTree(); size--; int *suffixArray =( int *) malloc ( sizeof ( int ) * size); buildSuffixArray(suffixArray); //Free the dynamically allocated memory freeSuffixTreeByPostOrder(root); free (suffixArray); strcpy (text, "GEEKSFORGEEKS$" ); buildSuffixTree(); size--; suffixArray =( int *) malloc ( sizeof ( int ) * size); buildSuffixArray(suffixArray); //Free the dynamically allocated memory freeSuffixTreeByPostOrder(root); free (suffixArray); strcpy (text, "AAAAAAAAAA$" ); buildSuffixTree(); size--; suffixArray =( int *) malloc ( sizeof ( int ) * size); buildSuffixArray(suffixArray); //Free the dynamically allocated memory freeSuffixTreeByPostOrder(root); free (suffixArray); strcpy (text, "ABCDEFG$" ); buildSuffixTree(); size--; suffixArray =( int *) malloc ( sizeof ( int ) * size); buildSuffixArray(suffixArray); //Free the dynamically allocated memory freeSuffixTreeByPostOrder(root); free (suffixArray); strcpy (text, "ABABABA$" ); buildSuffixTree(); size--; suffixArray =( int *) malloc ( sizeof ( int ) * size); buildSuffixArray(suffixArray); //Free the dynamically allocated memory freeSuffixTreeByPostOrder(root); free (suffixArray); strcpy (text, "abcabxabcd$" ); buildSuffixTree(); size--; suffixArray =( int *) malloc ( sizeof ( int ) * size); buildSuffixArray(suffixArray); //Free the dynamically allocated memory freeSuffixTreeByPostOrder(root); free (suffixArray); strcpy (text, "CCAAACCCGATTA$" ); buildSuffixTree(); size--; suffixArray =( int *) malloc ( sizeof ( int ) * size); buildSuffixArray(suffixArray); //Free the dynamically allocated memory freeSuffixTreeByPostOrder(root); free (suffixArray); return 0; } |
输出：
Suffix Array for String banana is: 5 3 1 0 4 2 Suffix Array for String GEEKSFORGEEKS is: 9 1 10 2 5 8 0 11 3 6 7 12 4 Suffix Array for String AAAAAAAAAA is: 9 8 7 6 5 4 3 2 1 0 Suffix Array for String ABCDEFG is: 0 1 2 3 4 5 6 Suffix Array for String ABABABA is: 6 4 2 0 5 3 1 Suffix Array for String abcabxabcd is: 0 6 3 1 7 4 2 8 9 5Suffix Array for String CCAAACCCGATTA is: 12 2 3 4 9 1 0 5 6 7 8 11 10
Ukkonen的后缀树构造需要O（N）时间和空间来为长度为N的字符串构建后缀树，然后遍历树需要O（N）来构建后缀数组。 所以总的来说，它在时间和空间上是线性的。 你能明白为什么遍历是O（N）吗？？因为长度为N的字符串的后缀树最多有N-1个内部节点和N个叶子。这些节点的遍历可以在O（N）中完成。 我们发表了以下更多关于后缀树应用程序的文章：
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