element distinctness, frequency moments, and sliding windows · previous element distinctness lower...
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![Page 1: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/1.jpg)
Element Distinctness, Frequency Moments, andSliding Windows
Raphael Clifford
University of Bristol, UK
arXiv:1309.3690
FOCS 2013
Joint work with Paul Beame and Widad Machmouchi
![Page 2: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/2.jpg)
Time-space tradeoffs
Wikipedia:Beer
1. Frequency moments. E.g. how many different beer cans?
2. Element distinctness (ED). Have I had the same can twice?
Particularly simple to solve if presorted.
![Page 3: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/3.jpg)
Time-space tradeoffs
What is the complexity of these problems using small space?
I Any solution using sorting requires T ∈ Ω(n2/S)[Borodin-Cook 82, Beame 91].
I What is the true complexity using small space?
I Are both problems really as hard as sorting? (No.)
I How about multi-output or sliding window versions?
![Page 4: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/4.jpg)
Time-space tradeoffs
What is the complexity of these problems using small space?
I Any solution using sorting requires T ∈ Ω(n2/S)[Borodin-Cook 82, Beame 91].
I What is the true complexity using small space?
I Are both problems really as hard as sorting? (No.)
I How about multi-output or sliding window versions?
![Page 5: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/5.jpg)
Time-space tradeoffs
What is the complexity of these problems using small space?
I Any solution using sorting requires T ∈ Ω(n2/S)[Borodin-Cook 82, Beame 91].
I What is the true complexity using small space?
I Are both problems really as hard as sorting? (No.)
I How about multi-output or sliding window versions?
![Page 6: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/6.jpg)
Time-space tradeoffs
What is the complexity of these problems using small space?
I Any solution using sorting requires T ∈ Ω(n2/S)[Borodin-Cook 82, Beame 91].
I What is the true complexity using small space?
I Are both problems really as hard as sorting? (No.)
I How about multi-output or sliding window versions?
![Page 7: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/7.jpg)
Sliding window ED and frequency moments
A B C B A B C
I #distinct elements (F0) = 3, 2, 3, 2, 3.
I Sliding window ED gives 1, 0, 1, 0, 1.
I We show sliding window ED is easier than sorting but slidingwindow F0 mod 2 is as hard as sorting.
![Page 8: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/8.jpg)
Our new results
Our new upper and lower bounds:
Single window Sliding window
Frequency moments T ∈ Ω(n√
log(n/S)/ log log(n/S)) [BSSV 03]T ∈ Ω(n2/S) (New)T ∈ O(n2/S) (New)
Element distinctnessT ∈ Ω(n
√log(n/S)/ log log(n/S)) [BSSV 03]
T ∈ O(n√n/S) (New)
T ∈ O(n√n/S) (New)
F0 mod 2 T ∈ O(n2/S) [PR 98]T ∈ O(n2/S) (New)T ∈ Ω(n2/S) (New)
Previous element distinctness lower bounds:
Comparison model Multi-way branching
Borodin et al. 1987 T ∈ Ω(n3/2√
log n/S) -
Yao 1988 T ∈ Ω(n2−ε(n)/S) -Ajtai 1999 - S ∈ o(n)⇒ T ∈ ω(n)
Beame et al. 2003 - T ∈ Ω(n√
log(n/S)/ log log(n/S))
![Page 9: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/9.jpg)
Our new results
Our new upper and lower bounds:
Single window Sliding window
Frequency moments T ∈ Ω(n√
log(n/S)/ log log(n/S)) [BSSV 03]T ∈ Ω(n2/S) (New)T ∈ O(n2/S) (New)
Element distinctnessT ∈ Ω(n
√log(n/S)/ log log(n/S)) [BSSV 03]
T ∈ O(n√n/S) (New)
T ∈ O(n√n/S) (New)
F0 mod 2 T ∈ O(n2/S) [PR 98]T ∈ O(n2/S) (New)T ∈ Ω(n2/S) (New)
Previous element distinctness lower bounds:
Comparison model Multi-way branching
Borodin et al. 1987 T ∈ Ω(n3/2√
log n/S) -
Yao 1988 T ∈ Ω(n2−ε(n)/S) -Ajtai 1999 - S ∈ o(n)⇒ T ∈ ω(n)
Beame et al. 2003 - T ∈ Ω(n√
log(n/S)/ log log(n/S))
![Page 10: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/10.jpg)
Our new results
Our new upper and lower bounds:
Single window Sliding window
Frequency moments T ∈ Ω(n√
log(n/S)/ log log(n/S)) [BSSV 03]T ∈ Ω(n2/S) (New)T ∈ O(n2/S) (New)
Element distinctnessT ∈ Ω(n
√log(n/S)/ log log(n/S)) [BSSV 03]
T ∈ O(n√n/S) (New)
T ∈ O(n√n/S) (New)
F0 mod 2 T ∈ O(n2/S) [PR 98]T ∈ O(n2/S) (New)T ∈ Ω(n2/S) (New)
Previous element distinctness lower bounds:
Comparison model Multi-way branching
Borodin et al. 1987 T ∈ Ω(n3/2√
log n/S) -
Yao 1988 T ∈ Ω(n2−ε(n)/S) -Ajtai 1999 - S ∈ o(n)⇒ T ∈ ω(n)
Beame et al. 2003 - T ∈ Ω(n√
log(n/S)/ log log(n/S))
![Page 11: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/11.jpg)
The overall method for our upper bounds
I Using Floyd’s (Pollard’s rho) cycle finding algorithm, we
construct a T ∈ O(n√
n/S) randomised branching programalgorithm for single window ED.
I 1-sided error, inversely polynomial in n
I Reduction from sliding-window ED to single window ED.I T ∈ O(n
√n/S) for a single window gives T ∈ O(n
√n/S) for
sliding windows
I Sliding window frequency moments T ∈ O(n2/S) in thecomparison model.
![Page 12: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/12.jpg)
Cycle finding in graphs
Floyd’s “tortoise and hare” algorithm.
Tortoise
Hare
![Page 13: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/13.jpg)
Cycle finding in graphs
Floyd’s “tortoise and hare” algorithm.
Tortoise
Hare
![Page 14: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/14.jpg)
Cycle finding in graphs
Floyd’s “tortoise and hare” algorithm.
Tortoise
Hare
![Page 15: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/15.jpg)
Cycle finding in graphs
Floyd’s “tortoise and hare” algorithm.
Tortoise
Hare
![Page 16: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/16.jpg)
Cycle finding in graphs
Floyd’s “tortoise and hare” algorithm.
Tortoise
Hare
![Page 17: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/17.jpg)
Cycle finding in graphs
Floyd’s “tortoise and hare” algorithm.
Tortoise
Hare
![Page 18: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/18.jpg)
Cycle finding in graphs
Floyd’s “tortoise and hare” algorithm.
Tortoise
Hare
![Page 19: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/19.jpg)
Cycle finding in graphs
Floyd’s “tortoise and hare” algorithm.
Tortoise
Hare
![Page 20: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/20.jpg)
Randomised cycle finding
134 921 37 812 396 452 921 98
1 2 3 4 5 6 7 8 9
157
ED input
![Page 21: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/21.jpg)
Randomised cycle finding
Random
Hash function
: → []
134 921 37 812 396 452 921 98
1 2 3 4 5 6 7 8 9
…
2…
6…
3…
5…
7…
3…
1…
…4
157
37
98
134
157
396
452
812
921
ED input
![Page 22: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/22.jpg)
Randomised cycle finding
Random
Hash function
: → []
134 921 37 812 396 452 921 98
1 2 3 4 5 6 7 8 9
…
2…
6…
3…
5…
7…
3…
1…
…4
157
37
98
134
157
396
452
812
921
ED input
![Page 23: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/23.jpg)
Randomised cycle finding
Random
Hash function
: → []
134 921 37 812 396 452 921 98
1 2 3 4 5 6 7 8 9
…
2…
6…
3…
5…
7…
3…
1…
…4
157
37
98
134
157
396
452
812
921
ED input
475
2
1
3
8 6
9
Induced Graph
![Page 24: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/24.jpg)
A new small-space upper bound for ED
Sampling uniformly from [n]:
I Expect to find a repeated value after Θ(√n) samples.
I Prob. of any fixed pair of numbers appearing before arepeated value approaches 2/n.
There is a reasonable chance of finding a real input duplicate inone cycle. Using constant space we can repeat Θ(n) times.
But to run faster using more space, we can’t just run S instancesin parallel.
![Page 25: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/25.jpg)
A new small-space upper bound for ED
Sampling uniformly from [n]:
I Expect to find a repeated value after Θ(√n) samples.
I Prob. of any fixed pair of numbers appearing before arepeated value approaches 2/n.
There is a reasonable chance of finding a real input duplicate inone cycle. Using constant space we can repeat Θ(n) times.
But to run faster using more space, we can’t just run S instancesin parallel.
![Page 26: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/26.jpg)
A new small-space upper bound for ED
I Maintain a redirection list to split cycles. Update listwhenever a new collision is found.
I Cycles roughly halve in length each time they are visited.
We find all collisions reachable from any S distinct starting pointsusing O(S) items of space and time roughly proportional to thesize of the subgraph explored.
![Page 27: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/27.jpg)
A new small-space upper bound for ED
Theorem
There is a randomised branching program algorithm computing EDwith 1-sided error that uses space S and T ∈ O(n
√n/S).
1. Run roughly n/S independent runs of collision-finding withindependent random choices of hash functions andindependent choices of roughly S starting indices.
2. Use run-time cut-off bounding the number of explored verticesat 2√Sn.
3. On each run, check if any of the collisions found is a duplicatein x , in which case output ED(x) = 0 and halt.
4. If none is found in any round then output ED(x) = 1.
![Page 28: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/28.jpg)
A new small-space upper bound for ED
Theorem
There is a randomised branching program algorithm computing EDwith 1-sided error that uses space S and T ∈ O(n
√n/S).
1. Run roughly n/S independent runs of collision-finding withindependent random choices of hash functions andindependent choices of roughly S starting indices.
2. Use run-time cut-off bounding the number of explored verticesat 2√Sn.
3. On each run, check if any of the collisions found is a duplicatein x , in which case output ED(x) = 0 and halt.
4. If none is found in any round then output ED(x) = 1.
![Page 29: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/29.jpg)
Sliding window ED
Theorem
Sliding window ED can be solved in time T ∈ O(n√n/S) with
1-sided error probability o(1/n).
Idea:
I Reduce to single window ED.
I A duplicate in one window determines a large number ofoutputs.
![Page 30: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/30.jpg)
A general sequential lower bound for sliding window F0
Framework of [Borodin-Cook 82, Abrahamson 91]
T ∈ Ω(n2/S) follows if, for some random input distribution, nomatter how cn input values are fixed, any fixed set of Θ(S) outputvalues occurs with prob. 2−S .
I Informally, for the first S outputs, it is hard to predict thenext output value even when a constant fraction of the inputvector is known.
![Page 31: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/31.jpg)
A general sequential lower bound for sliding window F0
Framework of [Borodin-Cook 82, Abrahamson 91]
T ∈ Ω(n2/S) follows if, for some random input distribution, nomatter how cn input values are fixed, any fixed set of Θ(S) outputvalues occurs with prob. 2−S .
I Informally, for the first S outputs, it is hard to predict thenext output value even when a constant fraction of the inputvector is known.
![Page 32: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/32.jpg)
The sliding window F0 lower bound
Need to show: for the first S outputs, it is hard to predict the nextoutput value even when a constant fraction of the input vector isknown.
I Input uniform over [n]2n−1. But some outputs are easy topredict.
a ? ? ? ? a
I Uniform distribution gives whp Ω(n) positions where:I xi is unique in the input.I F0(i , i + n − 1) of the window is in the range [0.5n, 0.85n].
I Only consider outputs for these positions and show they arehard to predict.
![Page 33: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/33.jpg)
The sliding window F0 lower bound
Need to show: for the first S outputs, it is hard to predict the nextoutput value even when a constant fraction of the input vector isknown.
I Input uniform over [n]2n−1. But some outputs are easy topredict.
a a a a a ?
I Uniform distribution gives whp Ω(n) positions where:I xi is unique in the input.I F0(i , i + n − 1) of the window is in the range [0.5n, 0.85n].
I Only consider outputs for these positions and show they arehard to predict.
![Page 34: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/34.jpg)
The sliding window F0 lower bound
Need to show: for the first S outputs, it is hard to predict the nextoutput value even when a constant fraction of the input vector isknown.
I Input uniform over [n]2n−1. But some outputs are easy topredict.
a b c d e ?
I Uniform distribution gives whp Ω(n) positions where:I xi is unique in the input.I F0(i , i + n − 1) of the window is in the range [0.5n, 0.85n].
I Only consider outputs for these positions and show they arehard to predict.
![Page 35: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/35.jpg)
The sliding window F0 lower bound
Need to show: for the first S outputs, it is hard to predict the nextoutput value even when a constant fraction of the input vector isknown.
I Input uniform over [n]2n−1. But some outputs are easy topredict.
a b c d e ?
I Uniform distribution gives whp Ω(n) positions where:I xi is unique in the input.I F0(i , i + n − 1) of the window is in the range [0.5n, 0.85n].
I Only consider outputs for these positions and show they arehard to predict.
![Page 36: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/36.jpg)
The sliding window F0 lower bound
Need to show: for the first S outputs, it is hard to predict the nextoutput value even when a constant fraction of the input vector isknown.
I Input uniform over [n]2n−1. But some outputs are easy topredict.
a b c d e ?
I Uniform distribution gives whp Ω(n) positions where:I xi is unique in the input.I F0(i , i + n − 1) of the window is in the range [0.5n, 0.85n].
I Only consider outputs for these positions and show they arehard to predict.
![Page 37: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/37.jpg)
Summing up
I Finding two identical items is easier than sorting the input.
I Sliding window element distinctness is still easier than sorting
I F0 mod 2 may be better than ED as an example of a harddecision problem to study.
I Sliding window F0 mod 2 has the same complexity as sorting(ignoring log factors).
I Is our new complexity for element distinctness in fact tight?
![Page 38: Element Distinctness, Frequency Moments, and Sliding Windows · Previous element distinctness lower bounds: Comparison model Multi-way branching Borodin et al. 1987 T 2 ( n3=2 p log](https://reader034.vdocuments.us/reader034/viewer/2022051912/60037caa77b4577e6807af2c/html5/thumbnails/38.jpg)
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