There is a well known problem called "Triple Step" that states:
"A child is running up a staircase with n steps and can hop either 1 step, 2 steps, or 3 steps at a time. Implement a method to count how many possible ways the child can run up the stairs"
The algorithm below is a version without memoization:
int countWays(int n) {
if (n < 0) {
return 0;
} else if (n == 0) {
return 1;
} else {
return countWays(n-1) + countWays(n-2) + countWays(n-3);
}
}
I know that its runtime can be improved from the exponential time complexity.
But I really would like to know how to build a dynamic programming table over this problem,
for example I tried the table below for n being 4 steps:
0 | 1 | 2 | 3 | 4 <= staircase size
1 1 | 1 | 1 | 1 | 1
2 1 | 1 | 2 | 2 | 3
3 1 | 1 | 2 | 3 | 4 <=** There's something wrong because for n=4 the output should be 7
Could someone give me a hint about how this table could be built for the problem above? (or maybe the table is fine and I'm not able to interpret it right)
Thanks!
As you mentioned that countWays(N) can be solved by taking sum of countWays(N-1), countWays(N-2) and countWays(N-3).
Since we know the answer for n<=0, we can start constructing our solution from n=0 to n=N and at any point of time we will always have N-1, N-2 and N-3 values ready to be used.
In the process of constructing solution from n=0 to n=N at any point of time we should have results of our earlier calculations stored somewhere.
you can take 3 variables to store these values and keep updating these 3 variables at each iteration to store the last 3 calculations.
int countWays(int n) {
int last = 1; // for n = 0
int secondLast = 0; // for n = -1
int thridLast = 0; // for n = -2
for(int i = 1 ; i <= n ; i++) {
int current = last + secondLast + thirdLast;
thirdLast = secondLast;
secondLast = last;
last = current;
}
return last;
}
instead of taking 3 variables you can store all the earlier calculations in an array and the code will look like this,
int countWays(int n) {
if(n<0) return 0;
int[] a = new int[n+3];
a[0] = 0;
a[1] = 0;
a[2] = 1; // stores the result for N=0
for(int i = 3 ; i < n+3 ; i++) {
a[i] = a[i-1] + a[i-2] + a[i-3];
}
return a[n+2];
}
and array will look like,
Answer -> [0, 0, 1, 1, 2, 4, 7]
Value Of N -> -2, -1,0 ,1, 2, 3, 4
Array created in this solutions is known is dynamic programming table also known as memoization or bottom up approach to DP
Run time complexity of above solution is O(N)
There is another way to solve these type of problems in O(Log N) time complexity, where solution can be described in terms of a linear recurrence relation.
The solution is known as Matrix Exponentiation, follow this link for more explanation - https://discuss.codechef.com/t/building-up-the-recurrence-matrix-to-compute-recurrences-in-o-logn-time/570
The table for this is 1d which is the staircase size, on every step x you add x-1, x-2, and x-3 if possible, for example:
0 | 1 | 2 | 3 | 4 <= staircase size
1st step 1 | 1 | 0 | 0 | 0 only x-1 is possible
2nd step 1 | 1 | 2 | 0 | 0 x-1 + x-2 are possible
3rd step 1 | 1 | 2 | 4 | 0 x-1 + x-2 + x-3 are possible
4th step 1 | 1 | 2 | 4 | 7 x-1 + x-2 + x-3 are possible
More explanation:
Step 1:
only reachable by 1-step
Step 2:
1-step + 1-step
2-steps
Step 3:
1-step + 1-step + 1-step
1-step + 2-steps
2-steps + 1-step
3-steps
Step 4:
1-step + 1-step + 1-step + 1-step
2-steps + 1-step + 1-step
1-step + 2-steps + 1-step
1-step + 1-step + 2-steps
1-step + 3-steps
3-steps + 1-step
2-steps + 2-steps
Assume that you are using dynamic programming to solve the problem.
Let a be the name of the variable of your table.
The formulae for a[n] with n = 0, 1, 2, .... are as you mentioned:
a[0] = 1
a[n] = a[n-1] + a[n-2] + a[n-3]
Be sure that a[n] for n < 0 is 0 always.
The answer for staircase size = 4 can be solved only if all the answers for 0 <= staircase size < 4 are given. i.e., a[4] can be calculated only if a[0], a[1], ..., a[3] are calculated.
The answer for staircase size = 3 can be solved only if all the answers for 0 <= staircase size < 3 are given. i.e., a[3] can be calculated only if a[0], ..., a[2] are calculated.
The answer for staircase size = 2 can be solved only if all the answers for 0 <= staircase size < 2 are given. i.e., a[2] can be calculated only if a[0], a[1] are calculated.
The answer for staircase size = 1 can be solved only if all the answers for 0 <= staircase size < 1 are given. i.e., a[1] can be calculated only if a[0] is calculated.
a[0] is the first formula.
Here, you can start.
a[0] = 1 // Initialization
a[1] = a[0] + a[-1] + a[-2] = a[0] + 0 + 0 // calculated at 1st loop (a[1] = 1)
a[2] = a[1] + a[0] + a[-1] = a[1] + a[0] + 0 // calculated at 2nd loop (a[2] = 1 + 1)
a[3] = a[2] + a[1] + a[0] // calculated at 3rd loop (a[3] = 2 + 1 + 1)
a[4] = a[3] + a[2] + a[1] // calculated at 4th loop (a[4] = 4 + 2 + 1)
...
a[n] = a[n-1] + a[n-2] + a[n-3] // calculated at nth loop
Related
I'm looking for a linear programming equation that satisfied the conditions;
Given that all variables here are binary variables
if A+B = 2; then C = 1; else C = 0
Also,
if A+B+D = 3; then E = 1; else E = 0
How would one phrase this and satisfy these conditions as well as linearity conditions?
I've tried
A + B - 2 <= M(1-y) and 1 - C <= My
for the first constraint but it doesn't seem to work
For the first equation, you can use:
C + 1 >= A + B
2C <= A + B
If there is a natural sense (max/min) for C in the problem, one of those is sufficient.
Similarly for the second:
E + 2 >= A + B + D
3E <= A + B + D
I have seen different solutions to the same problem, but none of them seem to use the approach I used. So here I'm trying to solve the classical coin-change problem in a bottom up dynamic programming approach using a DP table.
int[][] dp = new int[nums.length+1][target+1];
for (int i = 0; i <= nums.length; ++i)
dp[i][0] = 1;
for (int i = 1; i < dp.length; ++i) {
for (int j = 1; j < dp[i].length; ++j) {
if (nums[i-1] <= j)
dp[i][j] = dp[i-1][j] + dp[i][j-nums[i-1]];
else
dp[i][j] = dp[i-1][j];
}
}
The above code generates the table. For fun if I have: {2,3,5} coins, and want to exchange 8, the table would look like:
1 0 0 0 0 0 0 0 0
1 0 1 0 1 0 1 0 1
1 0 1 1 1 1 2 1 2
1 0 1 1 1 2 2 2 3
For the above the following method seem to be working well:
current <- 4
while (current > 0) do
i <- current
j <- 8
while (j > 0) do
if (dp[i][j] != dp[i-1][j]) then
nums[i-1] is part of the current solution
j <- j-1
else
i <- i-1
end
end
current <- current-1
end
Walking through for the above example, we get the following solutions:
1) [5,3]
2) [3,3,2]
3) [2,2,2,2]
Which is great! At least I thought, until I tried: {1,2} with a T=4. For this the table looks like:
1 0 0 0 0
1 1 1 1 1
1 1 2 2 3
With the above algorithm to print all solutions, I only get:
[2,2]
[1,1,1,1]
Which means I won't recover [2,1,1]. So this question is not about the generic how to print the solutions for different approaches to this problem, but how can I read the above DP table to find all solutions.
Well I have one solution but sure... I can see why the other similar answers are using different approaches for this problem. So the algorithm to generate all the possible answers from the above table looks like:
private List<List<Integer>> readSolutions(int[] nums, int[][] dp, int currentRow, int currentColumn, List<Integer> partialSolution) {
if (currentRow == 0) {
return new ArrayList<>();
}
if (currentColumn == 0) {
return new ArrayList<List<Integer>>(Collections.singleton(partialSolution));
}
List<List<Integer>> solutions = new ArrayList<>();
if (dp[currentRow][currentColumn] != dp[currentRow-1][currentColumn]) {
List<Integer> newSolution = new ArrayList<>(partialSolution);
newSolution.add(nums[currentRow-1]);
solutions.addAll(readSolutions(nums, dp, currentRow, currentColumn-nums[currentRow-1], newSolution));
solutions.addAll(readSolutions(nums, dp, currentRow-1, currentColumn, partialSolution));
return solutions;
}
return readSolutions(nums, dp, currentRow-1, currentColumn, partialSolution);
}
The logic on the other hand is simple. Take the above table for example:
0 1 2 3 4
0 1 0 0 0 0
1 1 1 1 1 1
2 1 1 2 2 3
To get a single solution we start from the bottom right corner. If the value does match with the value directly above us, we move up. If it doesn't we move left by the amount corresponding to the row we're in. To generate all answers on on the other hand from the above table...
we're at some position (i,j)
if the value at (i-1,j) is the same as (i,j) we make a recursive call to (i-1,j)
if the values do not match, we have 2 choices...
we can use the number corresponding to the current row, and recurse into (i,j-n)
we can skip the number and check if we can create (i,j) instead by using a recursive call to (i-1,j) instead.
if we reach the first row, we return an empty list
if we reach the first column, we return what we have already found, which will have the sum of target.
Looks awful right, but works as expected.
I have a 5x-5 maze specified as follows.
r = [1 0 1 1 1
1 1 1 0 1
0 1 0 0 1
1 1 1 0 1
1 0 1 0 1];
Where 1's are the paths and 0's are the walls.
Assume I have a function foo(policy_vector, r) that maps the elements of the policy vector to the elements in r. For example 1=UP, 2=Right, 3=Down, 4=Left. The MDP is set up such that the wall states are never realized so policies for those states are ignored in the plot.
policy_vector' = [3 2 2 2 3 2 2 1 2 3 1 1 1 2 3 2 1 4 2 3 1 1 1 2 2]
symbols' = [v > > > v > > ^ > v ^ ^ ^ > v > ^ < > v ^ ^ ^ > >]
I am trying to display my policy decision for a Markov Decision Process in the context of solving a maze. How would I plot something that looks like this? Matlab is preferable but Python is fine.
Even if some body could show me how to make a plot like this I would be able to figure it out from there.
function[] = policy_plot(policy,r)
[row,col] = size(r);
symbols = {'^', '>', 'v', '<'};
policy_symbolic = get_policy_symbols(policy, symbols);
figure()
hold on
axis([0, row, 0, col])
grid on
cnt = 1;
fill([0,0,col,col],[row,0,0,row],'k')
for rr = row:-1:1
for cc = 1:col
if r(row+1 - rr,cc) ~= 0 && ~(row == row+1 - rr && col == cc)
fill([cc-1,cc-1,cc,cc],[rr,rr-1,rr-1,rr],'g')
text(cc - 0.55,rr - 0.5,policy_symbolic{cnt})
end
cnt = cnt + 1;
end
end
fill([cc-1,cc-1,cc,cc],[rr,rr-1,rr-1,rr],'b')
text(cc - 0.70,rr - 0.5,'Goal')
function [policy_symbolic] = get_policy_symbols(policy, symbols)
policy_symbolic = cell(size(policy));
for ii = 1:length(policy)
policy_symbolic{ii} = symbols{policy(ii)};
end
I have a string S which consists of a's and b's. Perform the below operation once. Objective is to obtain the lexicographically smallest string.
Operation: Reverse exactly one substring of S
e.g.
if S = abab then Output = aabb (reverse ba of string S)
if S = abba then Output = aabb (reverse bba of string S)
My approach
Case 1: If all characters of the input string are same then output will be the string itself.
Case 2: if S is of the form aaaaaaa....bbbbbb.... then answer will be S itself.
otherwise: Find the first occurence of b in S say the position is i. String S will look like
aa...bbb...aaaa...bbbb....aaaa....bbbb....aaaaa...
|
i
In order to obtain the lexicographically smallest string the substring that will be reversed starts from index i. See below for possible ending j.
aa...bbb...aaaa...bbbb....aaaa....bbbb....aaaaa...
| | | |
i j j j
Reverse substring S[i:j] for every j and find the smallest string.
The complexity of the algorithm will be O(|S|*|S|) where |S| is the length of the string.
Is there a better way to solve this problem? Probably O(|S|) solution.
What I am thinking if we can pick the correct j in linear time then we are done. We will pick that j where number of a's is maximum. If there is one maximum then we solved the problem but what if it's not the case? I have tried a lot. Please help.
So, I came up with an algorithm, that seems to be more efficient that O(|S|^2), but I'm not quite sure of it's complexity. Here's a rough outline:
Strip of the leading a's, storing in variable start.
Group the rest of the string into letter chunks.
Find the indices of the groups with the longest sequences of a's.
If only one index remains, proceed to 10.
Filter these indices so that the length of the [first] group of b's after reversal is at a minimum.
If only one index remains, proceed to 10.
Filter these indices so that the length of the [first] group of a's (not including the leading a's) after reversal is at a minimum.
If only one index remains, proceed to 10.
Go back to 5, except inspect the [second/third/...] groups of a's and b's this time.
Return start, plus the reversed groups up to index, plus the remaining groups.
Since any substring that is being reversed begins with a b and ends in an a, no two hypothesized reversals are palindromes and thus two reversals will not result in the same output, guaranteeing that there is a unique optimal solution and that the algorithm will terminate.
My intuition says this approach of probably O(log(|S|)*|S|), but I'm not too sure. An example implementation (not a very good one albeit) in Python is provided below.
from itertools import groupby
def get_next_bs(i, groups, off):
d = 1 + 2*off
before_bs = len(groups[i-d]) if i >= d else 0
after_bs = len(groups[i+d]) if i <= d and len(groups) > i + d else 0
return before_bs + after_bs
def get_next_as(i, groups, off):
d = 2*(off + 1)
return len(groups[d+1]) if i < d else len(groups[i-d])
def maximal_reversal(s):
# example input: 'aabaababbaababbaabbbaa'
first_b = s.find('b')
start, rest = s[:first_b], s[first_b:]
# 'aa', 'baababbaababbaabbbaa'
groups = [''.join(g) for _, g in groupby(rest)]
# ['b', 'aa', 'b', 'a', 'bb', 'aa', 'b', 'a', 'bb', 'aa', 'bbb', 'aa']
try:
max_length = max(len(g) for g in groups if g[0] == 'a')
except ValueError:
return s # no a's after the start, no reversal needed
indices = [i for i, g in enumerate(groups) if g[0] == 'a' and len(g) == max_length]
# [1, 5, 9, 11]
off = 0
while len(indices) > 1:
min_bs = min(get_next_bs(i, groups, off) for i in indices)
indices = [i for i in indices if get_next_bs(i, groups, off) == min_bs]
# off 0: [1, 5, 9], off 1: [5, 9], off 2: [9]
if len(indices) == 1:
break
max_as = max(get_next_as(i, groups, off) for i in indices)
indices = [i for i in indices if get_next_as(i, groups, off) == max_as]
# off 0: [1, 5, 9], off 1: [5, 9]
off += 1
i = indices[0]
groups[:i+1] = groups[:i+1][::-1]
return start + ''.join(groups)
# 'aaaabbabaabbabaabbbbaa'
TL;DR: Here's an algorithm that only iterates over the string once (with O(|S|)-ish complexity for limited string lengths). The example with which I explain it below is a bit long-winded, but the algorithm is really quite simple:
Iterate over the string, and update its value interpreted as a reverse (lsb-to-msb) binary number.
If you find the last zero of a sequence of zeros that is longer than the current maximum, store the current position, and the current reverse value. From then on, also update this value, interpreting the rest of the string as a forward (msb-to-lsb) binary number.
If you find the last zero of a sequence of zeros that is as long as the current maximum, compare the current reverse value with the current value of the stored end-point; if it is smaller, replace the end-point with the current position.
So you're basically comparing the value of the string if it were reversed up to the current point, with the value of the string if it were only reversed up to a (so-far) optimal point, and updating this optimal point on-the-fly.
Here's a quick code example; it could undoubtedly be coded more elegantly:
function reverseSubsequence(str) {
var reverse = 0, max = 0, first, last, value, len = 0, unit = 1;
for (var pos = 0; pos < str.length; pos++) {
var digit = str.charCodeAt(pos) - 97; // read next digit
if (digit == 0) {
if (first == undefined) continue; // skip leading zeros
if (++len > max || len == max && reverse < value) { // better endpoint found
max = len;
last = pos;
value = reverse;
}
} else {
if (first == undefined) first = pos; // end of leading zeros
len = 0;
}
reverse += unit * digit; // update reverse value
unit <<= 1;
value = value * 2 + digit; // update endpoint value
}
return {from: first || 0, to: last || 0};
}
var result = reverseSubsequence("aaabbaabaaabbabaaabaaab");
document.write(result.from + "→" + result.to);
(The code could be simplified by comparing reverse and value whenever a zero is found, and not just when the end of a maximally long sequence of zeros is encountered.)
You can create an algorithm that only iterates over the input once, and can process an incoming stream of unknown length, by keeping track of two values: the value of the whole string interpreted as a reverse (lsb-to-msb) binary number, and the value of the string with one part reversed. Whenever the reverse value goes below the value of the stored best end-point, a better end-point has been found.
Consider this string as an example:
aaabbaabaaabbabaaabaaab
or, written with zeros and ones for simplicity:
00011001000110100010001
We iterate over the leading zeros until we find the first one:
0001
^
This is the start of the sequence we'll want to reverse. We will start interpreting the stream of zeros and ones as a reversed (lsb-to-msb) binary number and update this number after every step:
reverse = 1, unit = 1
Then at every step, we double the unit and update the reverse number:
0001 reverse = 1
00011 unit = 2; reverse = 1 + 1 * 2 = 3
000110 unit = 4; reverse = 3 + 0 * 4 = 3
0001100 unit = 8; reverse = 3 + 0 * 8 = 3
At this point we find a one, and the sequence of zeros comes to an end. It contains 2 zeros, which is currently the maximum, so we store the current position as a possible end-point, and also store the current reverse value:
endpoint = {position = 6, value = 3}
Then we go on iterating over the string, but at every step, we update the value of the possible endpoint, but now as a normal (msb-to-lsb) binary number:
00011001 unit = 16; reverse = 3 + 1 * 16 = 19
endpoint.value *= 2 + 1 = 7
000110010 unit = 32; reverse = 19 + 0 * 32 = 19
endpoint.value *= 2 + 0 = 14
0001100100 unit = 64; reverse = 19 + 0 * 64 = 19
endpoint.value *= 2 + 0 = 28
00011001000 unit = 128; reverse = 19 + 0 * 128 = 19
endpoint.value *= 2 + 0 = 56
At this point we find that we have a sequence of 3 zeros, which is longer that the current maximum of 2, so we throw away the end-point we had so far and replace it with the current position and reverse value:
endpoint = {position = 10, value = 19}
And then we go on iterating over the string:
000110010001 unit = 256; reverse = 19 + 1 * 256 = 275
endpoint.value *= 2 + 1 = 39
0001100100011 unit = 512; reverse = 275 + 1 * 512 = 778
endpoint.value *= 2 + 1 = 79
00011001000110 unit = 1024; reverse = 778 + 0 * 1024 = 778
endpoint.value *= 2 + 0 = 158
000110010001101 unit = 2048; reverse = 778 + 1 * 2048 = 2826
endpoint.value *= 2 + 1 = 317
0001100100011010 unit = 4096; reverse = 2826 + 0 * 4096 = 2826
endpoint.value *= 2 + 0 = 634
00011001000110100 unit = 8192; reverse = 2826 + 0 * 8192 = 2826
endpoint.value *= 2 + 0 = 1268
000110010001101000 unit = 16384; reverse = 2826 + 0 * 16384 = 2826
endpoint.value *= 2 + 0 = 2536
Here we find that we have another sequence with 3 zeros, so we compare the current reverse value with the end-point's value, and find that the stored endpoint has a lower value:
endpoint.value = 2536 < reverse = 2826
so we keep the end-point set to position 10 and we go on iterating over the string:
0001100100011010001 unit = 32768; reverse = 2826 + 1 * 32768 = 35594
endpoint.value *= 2 + 1 = 5073
00011001000110100010 unit = 65536; reverse = 35594 + 0 * 65536 = 35594
endpoint.value *= 2 + 0 = 10146
000110010001101000100 unit = 131072; reverse = 35594 + 0 * 131072 = 35594
endpoint.value *= 2 + 0 = 20292
0001100100011010001000 unit = 262144; reverse = 35594 + 0 * 262144 = 35594
endpoint.value *= 2 + 0 = 40584
And we find another sequence of 3 zeros, so we compare this position to the stored end-point:
endpoint.value = 40584 > reverse = 35594
and we find it has a smaller value, so we replace the possible end-point with the current position:
endpoint = {position = 21, value = 35594}
And then we iterate over the final digit:
00011001000110100010001 unit = 524288; reverse = 35594 + 1 * 524288 = 559882
endpoint.value *= 2 + 1 = 71189
So at the end we find that position 21 gives us the lowest value, so it is the optimal solution:
00011001000110100010001 -> 00000010001011000100111
^ ^
start = 3 end = 21
Here's a C++ version that uses a vector of bool instead of integers. It can parse strings longer than 64 characters, but the complexity is probably quadratic.
#include <vector>
struct range {unsigned int first; unsigned int last;};
range lexiLeastRev(std::string const &str) {
unsigned int len = str.length(), first = 0, last = 0, run = 0, max_run = 0;
std::vector<bool> forward(0), reverse(0);
bool leading_zeros = true;
for (unsigned int pos = 0; pos < len; pos++) {
bool digit = str[pos] - 'a';
if (!digit) {
if (leading_zeros) continue;
if (++run > max_run || run == max_run && reverse < forward) {
max_run = run;
last = pos;
forward = reverse;
}
}
else {
if (leading_zeros) {
leading_zeros = false;
first = pos;
}
run = 0;
}
forward.push_back(digit);
reverse.insert(reverse.begin(), digit);
}
return range {first, last};
}
Given a string s containing only lower case alphabets (a - z), find (i.e print) the characters that are repeated.
For ex, if string s = "aabcacdddec"
Output: a c d
3 approaches to this problem exists:
[brute force] Check every char of string (i.e s[i] with every other char and print if both are same)
Time complexity: O(n^2)
Space complexity: O(1)
[sort and then compare adjacent elements] After sorting (in O(n log(n) time), traverse the string and check if s[i] ans s[i + 1] are equal
Time complexity: O(n logn) + O(n) = O(n logn)
Space complexity: O(1)
[store the character count in an array] Create an array of size 26 (to keep track of a - z) and for every s[i], increment value stored at index = s[i] - 26 in the array. Finally traverse the array and print all elements (i.e 'a' + i) with value greater than 1
Time complexity: O(n)
Space complexity: O(1) but we have a separate array for storing the frequency of each element.
Is there a O(n) approach that DOES NOT use any array/hash table/map (etc)?
HINT: Use BIT Vectors
This is the element distinctness problem, so generally speaking - no there is no way to solve it in O(n) without extra space.
However, if you regard the alphabet as constant size (a-z characters only is pretty constant) you can either create a bitset of these characters, in O(1) space [ it is constant!] or check for each character in O(n) if it repeats more than once, it will be O(constant*n), which is still in O(n).
Pseudo code for 1st solution:
bit seen[] = new bit[SIZE_OF_ALPHABET] //contant!
bit printed[] = new bit[SIZE_OF_ALPHABET] //so is this!
for each i in seen.length: //init:
seen[i] = 0
printed[i] = 0
for each character c in string: //traverse the string:
i = intValue(c)
//already seen it and didn't print it? print it now!
if seen[i] == 1 and printed[i] == 0:
print c
printed[i] = 1
else:
seen[i] = 1
Pseudo code for 2nd solution:
for each character c from a-z: //constant number of repeats is O(1)
count = 0
for each character x in the string: //O(n)
if x==c:
count += 1
if count > 1
print count
Implementation in Java
public static void findDuplicate(String str) {
int checker = 0;
char c = 'a';
for (int i = 0; i < str.length(); ++i) {
int val = str.charAt(i) - c;
if ((checker & (1 << val)) > 0) {
System.out.println((char)(c+val));
}else{
checker |= (1 << val);
}
}
}
Uses as int as storage and performs bit wise operator to find the duplicates.
it is in O(n) .. explanation follows
Input as "abddc"
i==0
STEP #1 : val = 98 - 98 (0) str.charAt(0) is a and conversion char to int is 98 ( ascii of 'a')
STEP #2 : 1 << val equal to ( 1 << 0 ) equal to 1 finally 1 & 0 is 0
STEP #3 : checker = 0 | ( 1 << 0) equal to 0 | 1 equal to 1 checker is 1
i==1
STEP #1 : val = 99 - 98 (1) str.charAt(1) is b and conversion char to int is 99 ( ascii of 'b')
STEP #2 : 1 << val equal to ( 1 << 1 ) equal to 2 finally 1 & 2 is 0
STEP #3 : checker = 2 | ( 1 << 1) equal to 2 | 1 equal to 2 finally checker is 2
i==2
STEP #1 : val = 101 - 98 (3) str.charAt(2) is d and conversion char to int is 101 ( ascii of 'd')
STEP #2 : 1 << val equal to ( 1 << 3 ) equal to 8 finally 2 & 8 is 0
STEP #3 : checker = 2 | ( 1 << 3) equal to 2 | 8 equal to 8 checker is 8
i==3
STEP #1 : val = 101 - 98 (3) str.charAt(3) is d and conversion char to int is 101 ( ascii of 'd')
STEP #2 : 1 << val equal to ( 1 << 3 ) equal to 8 finally 8 & 8 is 8
Now print 'd' since the value > 0
You can also use the Bit Vector, depends upon the language it would space efficient. In java i would prefer to use int for this fixed ( just 26) constant case
The size of the character set is a constant, so you could scan the input 26 times. All you need is a counter to store the number of times you've seen the character corresponding to the current iteration. At the end of each iteration, print that character if your counter is greater than 1.
It's O(n) in runtime and O(1) in auxiliary space.
Implementation in C# (recursive solution)
static void getNonUniqueElements(string s, string nonUnique)
{
if (s.Count() > 0)
{
char ch = s[0];
s = s.Substring(1);
if (s.LastIndexOf(ch) > 0)
{
if (nonUnique.LastIndexOf(ch) < 0)
nonUnique += ch;
}
getNonUniqueElements(s, nonUnique);
}
else
{
Console.WriteLine(nonUnique);
return;
}
}
static void Main(string[] args)
{
getNonUniqueElements("aabcacdddec", "");
Console.ReadKey();
}