Sum of zero sum series
up vote
0
down vote
favorite
Given $n$ and $k$, I would like to know how to compute$$sum_{substack{x_0 ⊕x_1⊕cdots⊕x_k=0\x_i≥0, 0≤i≤k\sumlimits_{i=0}^kx_i≤n-2k}}binom{n-k-sumlimits_{i=0}^kx_i}k$$ in $O(nk·log n)$ time, where $⊕$ is exclusive or.
combinatorics algorithms
edited Nov 24 at 6:20
Saad
19.7k92252
asked Nov 24 at 6:07
Hang Wu
404210
add a comment |
up vote
0
down vote
favorite
Given $n$ and $k$, I would like to know how to compute$$sum_{substack{x_0 ⊕x_1⊕cdots⊕x_k=0\x_i≥0, 0≤i≤k\sumlimits_{i=0}^kx_i≤n-2k}}binom{n-k-sumlimits_{i=0}^kx_i}k$$ in $O(nk·log n)$ time, where $⊕$ is exclusive or.
combinatorics algorithms
edited Nov 24 at 6:20
Saad
19.7k92252
asked Nov 24 at 6:07
Hang Wu
404210
This looks easy
– David Peterson
Nov 24 at 6:50
Could you please explain a little more?
– Hang Wu
Nov 24 at 9:15
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given $n$ and $k$, I would like to know how to compute$$sum_{substack{x_0 ⊕x_1⊕cdots⊕x_k=0\x_i≥0, 0≤i≤k\sumlimits_{i=0}^kx_i≤n-2k}}binom{n-k-sumlimits_{i=0}^kx_i}k$$ in $O(nk·log n)$ time, where $⊕$ is exclusive or.
combinatorics algorithms
edited Nov 24 at 6:20
Saad
19.7k92252
asked Nov 24 at 6:07
Hang Wu
404210
Given $n$ and $k$, I would like to know how to compute$$sum_{substack{x_0 ⊕x_1⊕cdots⊕x_k=0\x_i≥0, 0≤i≤k\sumlimits_{i=0}^kx_i≤n-2k}}binom{n-k-sumlimits_{i=0}^kx_i}k$$ in $O(nk·log n)$ time, where $⊕$ is exclusive or.
combinatorics algorithms
combinatorics algorithms
edited Nov 24 at 6:20
Saad
19.7k92252
asked Nov 24 at 6:07
Hang Wu
404210
edited Nov 24 at 6:20
Saad
19.7k92252
asked Nov 24 at 6:07
Hang Wu
404210
edited Nov 24 at 6:20
Saad
19.7k92252
edited Nov 24 at 6:20
Saad
19.7k92252
edited Nov 24 at 6:20
Saad
19.7k92252
19.7k92252
asked Nov 24 at 6:07
Hang Wu
404210
asked Nov 24 at 6:07
Hang Wu
404210
asked Nov 24 at 6:07
Hang Wu
404210
404210
This looks easy
– David Peterson
Nov 24 at 6:50
Could you please explain a little more?
– Hang Wu
Nov 24 at 9:15
add a comment |
This looks easy
– David Peterson
Nov 24 at 6:50
Could you please explain a little more?
– Hang Wu
Nov 24 at 9:15
This looks easy
– David Peterson
Nov 24 at 6:50
This looks easy
– David Peterson
Nov 24 at 6:50
Could you please explain a little more?
– Hang Wu
Nov 24 at 9:15
Could you please explain a little more?
– Hang Wu
Nov 24 at 9:15
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
I finally figure out an answer. This paper "Nim Fractals" by Tanya Khovanova and Joshua Xiong gives a recursive formula to count the solutions to $$begin{cases} x_0 oplus cdots oplus x_k =0 \sum_{i=0}^{k}x_i=Send{cases}$$ in $O(kS)$. Denote the number as $f(k,S)$. We may utilize $f$ to see that the problem reduces to $sum_{S=0}^{n-2k}binom{n-k-S}{k}f(k,S)$. The overall complexity is $O(nkcdot log(n))$.
answered Nov 29 at 4:32
Hang Wu
404210
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: "69"
};
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: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
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
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2fmath.stackexchange.com%2fquestions%2f3011248%2fsum-of-zero-sum-series%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
I finally figure out an answer. This paper "Nim Fractals" by Tanya Khovanova and Joshua Xiong gives a recursive formula to count the solutions to $$begin{cases} x_0 oplus cdots oplus x_k =0 \sum_{i=0}^{k}x_i=Send{cases}$$ in $O(kS)$. Denote the number as $f(k,S)$. We may utilize $f$ to see that the problem reduces to $sum_{S=0}^{n-2k}binom{n-k-S}{k}f(k,S)$. The overall complexity is $O(nkcdot log(n))$.
answered Nov 29 at 4:32
Hang Wu
404210
add a comment |
up vote
0
down vote
accepted
I finally figure out an answer. This paper "Nim Fractals" by Tanya Khovanova and Joshua Xiong gives a recursive formula to count the solutions to $$begin{cases} x_0 oplus cdots oplus x_k =0 \sum_{i=0}^{k}x_i=Send{cases}$$ in $O(kS)$. Denote the number as $f(k,S)$. We may utilize $f$ to see that the problem reduces to $sum_{S=0}^{n-2k}binom{n-k-S}{k}f(k,S)$. The overall complexity is $O(nkcdot log(n))$.
answered Nov 29 at 4:32
Hang Wu
404210
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
I finally figure out an answer. This paper "Nim Fractals" by Tanya Khovanova and Joshua Xiong gives a recursive formula to count the solutions to $$begin{cases} x_0 oplus cdots oplus x_k =0 \sum_{i=0}^{k}x_i=Send{cases}$$ in $O(kS)$. Denote the number as $f(k,S)$. We may utilize $f$ to see that the problem reduces to $sum_{S=0}^{n-2k}binom{n-k-S}{k}f(k,S)$. The overall complexity is $O(nkcdot log(n))$.
answered Nov 29 at 4:32
Hang Wu
404210
I finally figure out an answer. This paper "Nim Fractals" by Tanya Khovanova and Joshua Xiong gives a recursive formula to count the solutions to $$begin{cases} x_0 oplus cdots oplus x_k =0 \sum_{i=0}^{k}x_i=Send{cases}$$ in $O(kS)$. Denote the number as $f(k,S)$. We may utilize $f$ to see that the problem reduces to $sum_{S=0}^{n-2k}binom{n-k-S}{k}f(k,S)$. The overall complexity is $O(nkcdot log(n))$.
answered Nov 29 at 4:32
Hang Wu
404210
answered Nov 29 at 4:32
Hang Wu
404210
answered Nov 29 at 4:32
Hang Wu
404210
answered Nov 29 at 4:32
Hang Wu
404210
404210
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics 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%2fmath.stackexchange.com%2fquestions%2f3011248%2fsum-of-zero-sum-series%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
This looks easy
– David Peterson
Nov 24 at 6:50
Could you please explain a little more?
– Hang Wu
Nov 24 at 9:15