Strassen algorithm for matrix multiplication complexity analysis
I see everywhere that the recursive equation for the complexity of Strassen alg is:
$$T(n) = 7T(tfrac{n}{2})+O(n^2).$$ This is not so clear to me.
The parameter $n$ is supposed to be the size of the input, but it seems that here it is one dimension of a matrix while the input size is actually $n^2$.
Also, each matrix of the input is divided to 4 sub matrices so it seems the recursive equation should be $$T(n) = 7T(tfrac{n}{4}) + O(n).$$
algorithms complexity-theory divide-and-conquer matrix
New contributor
add a comment |
I see everywhere that the recursive equation for the complexity of Strassen alg is:
$$T(n) = 7T(tfrac{n}{2})+O(n^2).$$ This is not so clear to me.
The parameter $n$ is supposed to be the size of the input, but it seems that here it is one dimension of a matrix while the input size is actually $n^2$.
Also, each matrix of the input is divided to 4 sub matrices so it seems the recursive equation should be $$T(n) = 7T(tfrac{n}{4}) + O(n).$$
algorithms complexity-theory divide-and-conquer matrix
New contributor
add a comment |
I see everywhere that the recursive equation for the complexity of Strassen alg is:
$$T(n) = 7T(tfrac{n}{2})+O(n^2).$$ This is not so clear to me.
The parameter $n$ is supposed to be the size of the input, but it seems that here it is one dimension of a matrix while the input size is actually $n^2$.
Also, each matrix of the input is divided to 4 sub matrices so it seems the recursive equation should be $$T(n) = 7T(tfrac{n}{4}) + O(n).$$
algorithms complexity-theory divide-and-conquer matrix
New contributor
I see everywhere that the recursive equation for the complexity of Strassen alg is:
$$T(n) = 7T(tfrac{n}{2})+O(n^2).$$ This is not so clear to me.
The parameter $n$ is supposed to be the size of the input, but it seems that here it is one dimension of a matrix while the input size is actually $n^2$.
Also, each matrix of the input is divided to 4 sub matrices so it seems the recursive equation should be $$T(n) = 7T(tfrac{n}{4}) + O(n).$$
algorithms complexity-theory divide-and-conquer matrix
algorithms complexity-theory divide-and-conquer matrix
New contributor
New contributor
edited Dec 17 at 18:15
OmG
1,400412
1,400412
New contributor
asked Dec 16 at 17:22
dafnahaktana
1442
1442
New contributor
New contributor
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
It's true that the parameter $n$ usually denotes the size of the input, but this is not always the case. For square matrix multiplication, $n$ denotes the number of rows (or columns). For graphs, $n$ often denotes the number of vertices, and $m$ the number of edges. For algorithms on Boolean functions, $n$ denotes the number of inputs, though the truth table itself has size $2^n$. There are many other examples.
add a comment |
It's back to the size of the matrix. Suppose the original matrix is $ntimes n$. Hence we will consider $T(n)$ as a computation of two matrix with size of $ntimes n$. When we divide the original matrix to 4 part, size of each part is $frac{n}{2}times frac{n}{2}$. Hence, the computation cost of multiplication of two matrices with this size is $T(frac{n}{2})$.
add a comment |
Time complexity is often based on the input size, but it is not an absolute requirement. In this case, for the multiplication of n x n matrices, it seems most natural to count the number of operations based on n, not on the problem size n x n.
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "419"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
dafnahaktana is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f101638%2fstrassen-algorithm-for-matrix-multiplication-complexity-analysis%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
It's true that the parameter $n$ usually denotes the size of the input, but this is not always the case. For square matrix multiplication, $n$ denotes the number of rows (or columns). For graphs, $n$ often denotes the number of vertices, and $m$ the number of edges. For algorithms on Boolean functions, $n$ denotes the number of inputs, though the truth table itself has size $2^n$. There are many other examples.
add a comment |
It's true that the parameter $n$ usually denotes the size of the input, but this is not always the case. For square matrix multiplication, $n$ denotes the number of rows (or columns). For graphs, $n$ often denotes the number of vertices, and $m$ the number of edges. For algorithms on Boolean functions, $n$ denotes the number of inputs, though the truth table itself has size $2^n$. There are many other examples.
add a comment |
It's true that the parameter $n$ usually denotes the size of the input, but this is not always the case. For square matrix multiplication, $n$ denotes the number of rows (or columns). For graphs, $n$ often denotes the number of vertices, and $m$ the number of edges. For algorithms on Boolean functions, $n$ denotes the number of inputs, though the truth table itself has size $2^n$. There are many other examples.
It's true that the parameter $n$ usually denotes the size of the input, but this is not always the case. For square matrix multiplication, $n$ denotes the number of rows (or columns). For graphs, $n$ often denotes the number of vertices, and $m$ the number of edges. For algorithms on Boolean functions, $n$ denotes the number of inputs, though the truth table itself has size $2^n$. There are many other examples.
answered Dec 16 at 19:04
Yuval Filmus
189k12177340
189k12177340
add a comment |
add a comment |
It's back to the size of the matrix. Suppose the original matrix is $ntimes n$. Hence we will consider $T(n)$ as a computation of two matrix with size of $ntimes n$. When we divide the original matrix to 4 part, size of each part is $frac{n}{2}times frac{n}{2}$. Hence, the computation cost of multiplication of two matrices with this size is $T(frac{n}{2})$.
add a comment |
It's back to the size of the matrix. Suppose the original matrix is $ntimes n$. Hence we will consider $T(n)$ as a computation of two matrix with size of $ntimes n$. When we divide the original matrix to 4 part, size of each part is $frac{n}{2}times frac{n}{2}$. Hence, the computation cost of multiplication of two matrices with this size is $T(frac{n}{2})$.
add a comment |
It's back to the size of the matrix. Suppose the original matrix is $ntimes n$. Hence we will consider $T(n)$ as a computation of two matrix with size of $ntimes n$. When we divide the original matrix to 4 part, size of each part is $frac{n}{2}times frac{n}{2}$. Hence, the computation cost of multiplication of two matrices with this size is $T(frac{n}{2})$.
It's back to the size of the matrix. Suppose the original matrix is $ntimes n$. Hence we will consider $T(n)$ as a computation of two matrix with size of $ntimes n$. When we divide the original matrix to 4 part, size of each part is $frac{n}{2}times frac{n}{2}$. Hence, the computation cost of multiplication of two matrices with this size is $T(frac{n}{2})$.
answered Dec 16 at 17:55
OmG
1,400412
1,400412
add a comment |
add a comment |
Time complexity is often based on the input size, but it is not an absolute requirement. In this case, for the multiplication of n x n matrices, it seems most natural to count the number of operations based on n, not on the problem size n x n.
add a comment |
Time complexity is often based on the input size, but it is not an absolute requirement. In this case, for the multiplication of n x n matrices, it seems most natural to count the number of operations based on n, not on the problem size n x n.
add a comment |
Time complexity is often based on the input size, but it is not an absolute requirement. In this case, for the multiplication of n x n matrices, it seems most natural to count the number of operations based on n, not on the problem size n x n.
Time complexity is often based on the input size, but it is not an absolute requirement. In this case, for the multiplication of n x n matrices, it seems most natural to count the number of operations based on n, not on the problem size n x n.
answered Dec 17 at 21:04
gnasher729
10.1k1115
10.1k1115
add a comment |
add a comment |
dafnahaktana is a new contributor. Be nice, and check out our Code of Conduct.
dafnahaktana is a new contributor. Be nice, and check out our Code of Conduct.
dafnahaktana is a new contributor. Be nice, and check out our Code of Conduct.
dafnahaktana is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to Computer Science Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f101638%2fstrassen-algorithm-for-matrix-multiplication-complexity-analysis%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown