4242
4343<!-- 这里可写通用的实现逻辑 -->
4444
45+ ** 方法一:直接相乘**
46+ 47+ 我们可以直接按照矩阵乘法的定义,计算出结果矩阵中的每一个元素。
48+ 49+ 时间复杂度 $O(m \times n \times k),ドル空间复杂度 $O(m \times n)$。
50+ 51+ ** 方法二:预处理**
52+ 53+ 我们可以预处理出两个矩阵的稀疏表示,即 $g1[ i] $ 表示矩阵 $mat1$ 第 $i$ 行中所有非零元素的列下标和值,而 $g2[ i] $ 表示矩阵 $mat2$ 第 $i$ 行中所有非零元素的列下标和值。
54+ 55+ 接下来,我们遍历每一行 $i,ドル遍历 $g1[ i] $ 中的每一个元素 $(k, x),ドル遍历 $g2[ k] $ 中的每一个元素 $(j, y),ドル那么最终 $mat1[ i] [ k ] \times mat2[ k] [ j ] $ 就会对应到结果矩阵中的 $ans[ i] [ j ] ,ドル我们将所有的结果累加即可。
56+ 57+ 时间复杂度 $O(m \times n \times k),ドル空间复杂度 $O(m \times n)$。
58+ 4559<!-- tabs:start -->
4660
4761### ** Python3**
4862
4963<!-- 这里可写当前语言的特殊实现逻辑 -->
5064
51- 直接模拟。
52- 5365``` python
5466class Solution :
5567 def multiply (self mat1 : List[List[int ]], mat2 : List[List[int ]]) -> List[List[int ]]:
56- r1, c1, c2 = len (mat1), len (mat1[ 0 ] ), len (mat2[0 ])
57- res = [[0 ] * c2 for _ in range (r1 )]
58- for i in range (r1 ):
59- for j in range (c2 ):
60- for k in range (c1 ):
61- res [i][j] += mat1[i][k] * mat2[k][j]
62- return res
68+ m, n = len (mat1), len (mat2[0 ])
69+ ans = [[0 ] * n for _ in range (m )]
70+ for i in range (m ):
71+ for j in range (n ):
72+ for k in range (len (mat2) ):
73+ ans [i][j] += mat1[i][k] * mat2[k][j]
74+ return ans
6375```
6476
65- 用哈希表记录稀疏矩阵 mat1 中的非 0 值。
66- 6777``` python
6878class Solution :
6979 def multiply (self mat1 : List[List[int ]], mat2 : List[List[int ]]) -> List[List[int ]]:
70- r1, c1, c2 = len (mat1), len (mat1[0 ]), len (mat2[0 ])
71- res = [[0 ] * c2 for _ in range (r1)]
72- mp = defaultdict(list )
73- for i in range (r1):
74- for j in range (c1):
75- if mat1[i][j] != 0 :
76- mp[i].append(j)
77- for i in range (r1):
78- for j in range (c2):
79- for k in mp[i]:
80- res[i][j] += mat1[i][k] * mat2[k][j]
81- return res
80+ def f (mat : List[List[int ]]) -> List[List[int ]]:
81+ g = [[] for _ in range (len (mat))]
82+ for i, row in enumerate (mat):
83+ for j, x in enumerate (row):
84+ if x:
85+ g[i].append((j, x))
86+ return g
87+ 88+ g1 = f(mat1)
89+ g2 = f(mat2)
90+ m, n = len (mat1), len (mat2[0 ])
91+ ans = [[0 ] * n for _ in range (m)]
92+ for i in range (m):
93+ for k, x in g1[i]:
94+ for j, y in g2[k]:
95+ ans[i][j] += x * y
96+ return ans
8297```
8398
8499### ** Java**
@@ -88,26 +103,51 @@ class Solution:
88103``` java
89104class Solution {
90105 public int [][] multiply (int [][] mat1 , int [][] mat2 ) {
91- int r1 = mat1. length, c1 = mat1[0 ]. length, c2 = mat2[0 ]. length;
92- int [][] res = new int [r1][c2];
93- Map<Integer , List<Integer > > mp = new HashMap<> ();
94- for (int i = 0 ; i < r1; ++ i) {
95- for (int j = 0 ; j < c1; ++ j) {
96- if (mat1[i][j] != 0 ) {
97- mp. computeIfAbsent(i, k - > new ArrayList<> ()). add(j);
106+ int m = mat1. length, n = mat2[0 ]. length;
107+ int [][] ans = new int [m][n];
108+ for (int i = 0 ; i < m; ++ i) {
109+ for (int j = 0 ; j < n; ++ j) {
110+ for (int k = 0 ; k < mat2. length; ++ k) {
111+ ans[i][j] += mat1[i][k] * mat2[k][j];
98112 }
99113 }
100114 }
101- for (int i = 0 ; i < r1; ++ i) {
102- for (int j = 0 ; j < c2; ++ j) {
103- if (mp. containsKey(i)) {
104- for (int k : mp. get(i)) {
105- res[i][j] += mat1[i][k] * mat2[k][j];
106- }
115+ return ans;
116+ }
117+ }
118+ ```
119+ 120+ ``` java
121+ class Solution {
122+ public int [][] multiply (int [][] mat1 , int [][] mat2 ) {
123+ int m = mat1. length, n = mat2[0 ]. length;
124+ int [][] ans = new int [m][n];
125+ var g1 = f(mat1);
126+ var g2 = f(mat2);
127+ for (int i = 0 ; i < m; ++ i) {
128+ for (int [] p : g1[i]) {
129+ int k = p[0 ], x = p[1 ];
130+ for (int [] q : g2[k]) {
131+ int j = q[0 ], y = q[1 ];
132+ ans[i][j] += x * y;
107133 }
108134 }
109135 }
110- return res;
136+ return ans;
137+ }
138+ 139+ private List<int[]> [] f (int [][] mat ) {
140+ int m = mat. length, n = mat[0 ]. length;
141+ List<int[]> [] g = new List [m];
142+ Arrays . setAll(g, i - > new ArrayList<> ());
143+ for (int i = 0 ; i < m; ++ i) {
144+ for (int j = 0 ; j < n; ++ j) {
145+ if (mat[i][j] != 0 ) {
146+ g[i]. add(new int [] {j, mat[i][j]});
147+ }
148+ }
149+ }
150+ return g;
111151 }
112152}
113153```
@@ -118,20 +158,48 @@ class Solution {
118158class Solution {
119159public:
120160 vector<vector<int >> multiply(vector<vector<int >>& mat1, vector<vector<int >>& mat2) {
121- int r1 = mat1.size(), c1 = mat1[ 0] .size(), c2 = mat2[ 0] .size();
122- vector<vector<int >> res(r1, vector<int >(c2));
123- unordered_map<int, vector<int >> mp;
124- for (int i = 0; i < r1; ++i) {
125- for (int j = 0; j < c1; ++j) {
126- if (mat1[ i] [ j ] != 0) mp[ i] .push_back(j);
161+ int m = mat1.size(), n = mat2[ 0] .size();
162+ vector<vector<int >> ans(m, vector<int >(n));
163+ for (int i = 0; i < m; ++i) {
164+ for (int j = 0; j < n; ++j) {
165+ for (int k = 0; k < mat2.size(); ++k) {
166+ ans[ i] [ j ] += mat1[ i] [ k ] * mat2[ k] [ j ] ;
167+ }
127168 }
128169 }
129- for (int i = 0; i < r1; ++i) {
130- for (int j = 0; j < c2; ++j) {
131- for (int k : mp[ i] ) res[ i] [ j ] += mat1[ i] [ k ] * mat2[ k] [ j ] ;
170+ return ans;
171+ }
172+ };
173+ ```
174+
175+ ```cpp
176+ class Solution {
177+ public:
178+ vector<vector<int>> multiply(vector<vector<int>>& mat1, vector<vector<int>>& mat2) {
179+ int m = mat1.size(), n = mat2[0].size();
180+ vector<vector<int>> ans(m, vector<int>(n));
181+ auto g1 = f(mat1), g2 = f(mat2);
182+ for (int i = 0; i < m; ++i) {
183+ for (auto& [k, x] : g1[i]) {
184+ for (auto& [j, y] : g2[k]) {
185+ ans[i][j] += x * y;
186+ }
132187 }
133188 }
134- return res;
189+ return ans;
190+ }
191+
192+ vector<vector<pair<int, int>>> f(vector<vector<int>>& mat) {
193+ int m = mat.size(), n = mat[0].size();
194+ vector<vector<pair<int, int>>> g(m);
195+ for (int i = 0; i < m; ++i) {
196+ for (int j = 0; j < n; ++j) {
197+ if (mat[i][j]) {
198+ g[i].emplace_back(j, mat[i][j]);
199+ }
200+ }
201+ }
202+ return g;
135203 }
136204};
137205```
@@ -140,27 +208,99 @@ public:
140208
141209``` go
142210func multiply (mat1 [][]int , mat2 [][]int ) [][]int {
143- r1, c1, c2 := len(mat1), len(mat1[0] ), len(mat2[0])
144- res := make([][]int, r1 )
145- for i := range res {
146- res [i] = make([]int, c2 )
211+ m , n := len (mat1), len (mat2[0 ])
212+ ans := make ([][]int , m )
213+ for i := range ans {
214+ ans [i] = make ([]int , n )
147215 }
148- mp := make(map[int][]int)
149- for i := 0; i < r1; i++ {
150- for j := 0; j < c1; j++ {
151- if mat1[i][j] != 0 {
152- mp[i] = append(mp[i], j)
216+ for i := 0 ; i < m; i++ {
217+ for j := 0 ; j < n; j++ {
218+ for k := 0 ; k < len (mat2); k++ {
219+ ans[i][j] += mat1[i][k] * mat2[k][j]
153220 }
154221 }
155222 }
156- for i := 0; i < r1; i++ {
157- for j := 0; j < c2; j++ {
158- for _, k := range mp[i] {
159- res[i][j] += mat1[i][k] * mat2[k][j]
223+ return ans
224+ }
225+ ```
226+ 227+ ``` go
228+ func multiply (mat1 [][]int , mat2 [][]int ) [][]int {
229+ m , n := len (mat1), len (mat2[0 ])
230+ ans := make ([][]int , m)
231+ for i := range ans {
232+ ans[i] = make ([]int , n)
233+ }
234+ f := func (mat [][]int ) [][][2 ]int {
235+ m , n := len (mat), len (mat[0 ])
236+ g := make ([][][2 ]int , m)
237+ for i := range g {
238+ g[i] = make ([][2 ]int , 0 , n)
239+ for j := range mat[i] {
240+ if mat[i][j] != 0 {
241+ g[i] = append (g[i], [2 ]int {j, mat[i][j]})
242+ }
160243 }
161244 }
245+ return g
162246 }
163- return res
247+ g1 , g2 := f (mat1), f (mat2)
248+ for i := range g1 {
249+ for _ , p := range g1[i] {
250+ k , x := p[0 ], p[1 ]
251+ for _ , q := range g2[k] {
252+ j , y := q[0 ], q[1 ]
253+ ans[i][j] += x * y
254+ }
255+ }
256+ }
257+ return ans
258+ }
259+ ```
260+ 261+ ### ** TypeScript**
262+ 263+ ``` ts
264+ function multiply(mat1 : number [][], mat2 : number [][]): number [][] {
265+ const = [mat1 .length , mat2 [0 ].length ];
266+ const : number [][] = Array .from ({ length: m }, () => Array .from ({ length: n }, () => 0 ));
267+ for (let i = 0 ; i < m ; ++ i ) {
268+ for (let j = 0 ; j < n ; ++ j ) {
269+ for (let k = 0 ; k < mat2 .length ; ++ k ) {
270+ ans [i ][j ] += mat1 [i ][k ] * mat2 [k ][j ];
271+ }
272+ }
273+ }
274+ return ans ;
275+ }
276+ ```
277+ 278+ ``` ts
279+ function multiply(mat1 : number [][], mat2 : number [][]): number [][] {
280+ const = [mat1 .length , mat2 [0 ].length ];
281+ const : number [][] = Array .from ({ length: m }, () => Array .from ({ length: n }, () => 0 ));
282+ const = (mat : number [][]): number [][][] => {
283+ const = [mat .length , mat [0 ].length ];
284+ const : number [][][] = Array .from ({ length: m }, () => []);
285+ for (let i = 0 ; i < m ; ++ i ) {
286+ for (let j = 0 ; j < n ; ++ j ) {
287+ if (mat [i ][j ] !== 0 ) {
288+ ans [i ].push ([j , mat [i ][j ]]);
289+ }
290+ }
291+ }
292+ return ans ;
293+ };
294+ const = f (mat1 );
295+ const = f (mat2 );
296+ for (let i = 0 ; i < m ; ++ i ) {
297+ for (const of g1 [i ]) {
298+ for (const of g2 [k ]) {
299+ ans [i ][j ] += x * y ;
300+ }
301+ }
302+ }
303+ return ans ;
164304}
165305```
166306
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