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Questions tagged [summation]

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3 votes
2 answers
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Linearity property of summation applied to Big Theta notation (CLRS math background appendix)

Section A.1 of the Mathematical Appendix of the CLRS, the third edition, page 1146, contains the following formula stating linearity property of summation applied to $\Theta$ notation: $$ \sum_{k=1}^{...
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4 votes
1 answer
124 views

Maximum sum of values in a square grid (one in each row/ column)

this is my first post here so bare with me :). What i'm looking for is an algorithm that can find the maximum sum of values in a square grid under the restriction, that you can only pick 1 value from ...
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  • 41
2 votes
1 answer
18 views

sum of Boolean characters larger degree

I was curious if someone knew the answer/reference for the following. So it is well-known that if $S\in \{0,1\}^n$, then $$ \frac{1}{2^n}\sum_{x\in \{0,1\}^n} (-1)^{\langle S, x\rangle}=1 $$ if and ...
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1 vote
1 answer
48 views

Is there a formula for this summation $\sum\limits_{k=i}^\mathbb{N}{i}$?

Is there a formulae for this summation? I wonder this could be not a constant or to say that i * i? $\sum\limits_{k=i}^\mathbb{N}{i}$
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0 votes
1 answer
32 views

Factorial Formulae proof (from Algorithm Design Manual)

I'm going through Algorithm Design Manual and it didn't take long before I hit a proof I don't understand. Can anyone point me in the right direction? From the book: Problem: Prove that $\sum_{i=1}^n ...
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0 votes
1 answer
18 views

Efficiently Taking the Sums of all Pairs of Elements from a Set

I have a set of numbers and I want to find the sums of all possible pairs of elements. With the set $\{1, 2, 3\}$ for example, I would want $\{1+2, 1+3, 2+3\}$ as my answer. I could do that in n^2 ...
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  • 103
0 votes
1 answer
93 views

Understanding a summation notation. Sum(j=2 to n) j - 1

I have been reading analysis of insertion sort in the "Introduction to algorithms" and faced a problem with understanding a specific summation notation when the worst case occurs. I know how ...
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0 votes
1 answer
54 views

Asymptotic notation for summations

I am struggling to understand why this property of asymptotic notation is true
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2 votes
2 answers
109 views

Closed-form expression for the sums of the rows of the Trinomial triangle

There is a question in the second chapter of the book that I'm unable to solve, and unfortunately algorist.com does not provide a rigorous enough solution, or maybe I can't quite understand it. Here ...
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  • 125
0 votes
2 answers
76 views

Summation and big theta

Can anyone suggest a starter book or website where I can find information and examples of putting functions under summation signs into Big Theta. basic Arihmetic , geometric, Quadratic are fine , but ...
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  • 19
4 votes
2 answers
6k views

Sum of series (n/1) + (n/2) + (n/3) + (n/4) +…….+ (n/n)

I wonder if there is a formula to calculate the sum of n/1 + n/2 + n/3 + n/4 + ... + 1. (Integer division) The number n can be as large as 10^12, so a formula or a ...
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0 votes
2 answers
275 views

Estimate the running time of a while-for-while loop

i, sq ← 1, 1 while sq < n for j ← 1 to sq k ← 1 while k ≤ j k ← 2 ∗ k i ← i + 1 sq ← i ∗ i I have expressed the running ...
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2 votes
0 answers
84 views

How to parallelize a summation efficiently

Say I have an array a[1..n], and I want to output an array s[1..n] with s[i] = a[1]+...+a[i]. What is the best (or at least standard) way to do so in parallel? The way I can think of doing it, given m ...
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0 votes
0 answers
61 views

Reconstructing an Array via Time-Intensive Subset Queries

I am trying to design an algorithm for a problem, and the following is an auxiliary problem for which a good solution would imply a faster algorithm for the original problem. I am given access to an ...
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0 votes
3 answers
144 views

Find the smallest group of numbers with sum bigger then $X$

Given a list of numbers $S$ where $0 < s_i < 100$, find the minimum sum group of numbers with a sum bigger than $X$. Each number can be used multiple times. Ex: for $S = [3,4.1], X = 10$ the ...
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-1 votes
2 answers
79 views

What would be the complexity for this relation?

The function: for (int i=1; i<n²; i++) for (int j=i; j<n; j++) print(j) Putting it into a relation, I got: $C1 + \sum_{i=1}^{n^2}(C2+...
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  • 111
2 votes
1 answer
789 views

How to solve recurrence $T(n) = 5T(\frac{n}{2}) + n^2\lg^2 n$

I have tried solve the recurrence $T(n) = 5T(\frac{n}{2}) + n^2\lg^2 n$ using substitution. Apparently, it is exact for some $n$ and the order of the general solution can be found from this exact ...
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0 votes
1 answer
71 views

Calculate the number of iterations in unusual nested loop

I am trying to calculate the number of iterations of a sequence of nested loops of the form: \begin{equation} N = \sum_{j=0}^{j_T} \sum_{k=0}^{j} \sum_{l=0}^{k} \sum_{n=n_0}^{n_T} 1 \end{equation} ...
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1 vote
2 answers
553 views

Summation of asymptotic notation

How can we solve summation of asymptotic notations like given below: $$ \sum_{k=1}^{n-1} O(n). $$
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4 votes
2 answers
253 views

Why is $\sum_{i=1}^n O(i)$ not the same as $O(1)+O(2)+\dots+O(n)$?

The well-known textbook Introduction to Algorithms ("CLRS", 3rd edition, chapter 3.1) claims the following: $$ \sum_{i=1}^n O(i) $$ is not the same as (I'm not using DNE because the book explicitly ...
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