Sum of zero sum series
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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
user302797
19.3k92251
asked Nov 24 at 6:07
Hang Wu
377110
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
user302797
19.3k92251
asked Nov 24 at 6:07
Hang Wu
377110
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
user302797
19.3k92251
asked Nov 24 at 6:07
Hang Wu
377110
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
user302797
19.3k92251
asked Nov 24 at 6:07
Hang Wu
377110
edited Nov 24 at 6:20
user302797
19.3k92251
asked Nov 24 at 6:07
Hang Wu
377110
edited Nov 24 at 6:20
user302797
19.3k92251
edited Nov 24 at 6:20
user302797
19.3k92251
edited Nov 24 at 6:20
user302797
19.3k92251
19.3k92251
asked Nov 24 at 6:07
Hang Wu
377110
asked Nov 24 at 6:07
Hang Wu
377110
asked Nov 24 at 6:07
Hang Wu
377110
377110
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
377110
add a comment |
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
377110
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
377110
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
377110
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
377110
answered Nov 29 at 4:32
Hang Wu
377110
answered Nov 29 at 4:32
Hang Wu
377110
answered Nov 29 at 4:32
Hang Wu
377110
377110
add a comment |
add a comment |
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This looks easy
– David Peterson
Nov 24 at 6:50
Could you please explain a little more?
– Hang Wu
Nov 24 at 9:15