We will hold AtCoder Beginner Contest 188.

- Contest URL: https://atcoder.jp/contests/abc188
- Start Time: http://www.timeanddate.com/worldclock/fixedtime.html?iso=20210110T2100&p1=248
- Duration: 100 minutes
- Number of Tasks: 6
- Writer: kort0n, kyopro_friends, QCFium, tatyam, YoshikaMiyafuji
- Rated range: ~ 1999

The point values will be 100-200-300-400-500-600.

We are looking forward to your participation!

[After contest]

How to solve D?

First i calculated ans by taking service for each day(c[i])

then i calculated for how many days i will be paying x amount of cost using prefix sum technique(Prefix sum using maps).

Then working smartly i store cost for day intervals(like x to y) in a data structure.

Then simply calculated if using C yen from day x to day y is useful or not.

for better understanding see this code below https://atcoder.jp/contests/abc188/submissions/19339488

Nice Explanation

Instead of that initial C consideration for all days, if we try and add min(interval*c, interval*cost) to the answer, it gives WA. why is that? I dont get how they're different? WA AC

Sorry for the wrong link. lol. WA this is the right one.

I think in WA you are paying C yen more than 1 time for same days. Check in your input loop.

Thanks, i got it. It was because, mp[prev] (that is the cost) could be 1e14 and interval could be 1e9. So that caused an overflow. We can avoid this because c ia at most 1e9. So if we consider c instead of a huge sum, it won't overflow.

E was easier than D . How to solve D and F ?

To solve D, I use Sweep-Line algorithm

tell me where i could read about it

You can read about it in: CPH

for D I use coordinate compression,

I understood the sweep-line solution , Could you please explain how coordinate compression can be used to solve the problem ? Or share your submission ?

If it's similar to the one given here isn't it same as sweep-line ? We are adding and subtracting from index (which are mapped to larger values and thus it can be said that they are "compressed") but it's same as adding and subtracting in map and traversing through it.

D seems pretty standard iteration over all discrete point boundaries and since for each range the minimum answer is the same for each day, we can add its range * minimum answer for the range to the final answer.

For F, I came up with this solution : Decide how many multiplication operations one will use (iterate on it, because it's max around log2(y/x) +- 2, say 'i' is the pointer). Now if we use these many multiplications, subtract contribution of original value * 2^(i) from the final required value. Now take the absolute value of the difference. We now need to arrange subtraction and addition operations in the original multiplication operations so as to make difference = 0 while minimizing total operations. Here we an observe that if we move optimally, there are at max 2*i + 1 distinct values that will be assumed by the difference if we start reducing from the highest bit. You can understand it like this : We either just reach the particular value or just exceed it using our difference operations at every bit. It doesn't make sense to go too far from the given difference. We can find these by taking modulo of difference, say m, with (2^j) making m and (2^j) — m as the optimal values for all j from 0 to i. Now we can run a dynamic programming from these values to get the minimum answer for i as dp[difference] + i. Minimum over all i is the required answer.

How to solve problem D?

Push the start/end of each service into a priority queue sorted by day, go from lowest to highest, keep track of current sum of separate fees and you can calculate Ans += min(sum, prime fee) * elapsed days

But how to judge that there are multiple services at the same time?I think it may cause time limited error.But I think you have better way to solve this problem,could you tell me ? Thank you.

Put beginnings and ends into one array and sort by their time (keep track to which interval they belong to and whether they are start/end of the interval). Then scan along this array and compute the current price for one day. This price holds for the whole interval from the current processed event (beginning/end of interval) to the previous event. Of course, instead of this price, we can choose the price C (the whole subscription). So we add the chosen price multiplied by the time since the last event to the result.

how to solve F?

Solution to F was already there on Stackoverflow.

True, but (i) it didn't get many views, so it is unlikely that many people knew it, and (ii) it does not say a word about answer caching, and you should not get AC, because without it solution is $$$\mathcal{O}(y)$$$ instead of $$$\mathcal{O}(\log y)$$$.

F is an easier version of AGC 44 Problem A whose editorial discusses the top down approach along with memoization.

Implemented using the same concept

AGC 44 Problem A

ABC 188 Problem F

An easier version of F : 520B

Yes, I agree with that. It was a good problem.

how to solve D ??

Can someone tell me if what I'm doing for F here is right or wrong? I'm passing 48 out of 49 cases

What is your logic?

I was basically working with the closest differences of x with y if we keep multiplying x by 2. But its wrong, this would miss some cases

Try this one:

`1 47`

. The answer is`7 = (1x2+1)x2x2x2x2-1`

Thanks! I got my mistake

(deleted)

I am getting one test case wrong in D. My Code Please let me know that case.

I was also getting the same verdict which was due to the integer overflow. Try to look for that.

Got it, thanks.

Suppose N is given 1e5 and all the ranges are starting from 1 and ending at 1e9 with the value of 1e9.

Total number of days = 1e9

Value of cost each day = 1e9 * 1e5 = 1e14

So total cost = days * cost per day = 1e9 * 1e14 which is the reason for integer overflow

So what you should do-

take min(cost each day, c) whose max value can only be 1e9 which is the max value for c.

Now total cost = 1e9 * days = 1e9 * 1e9 = 1e18(max)

Thanks a lot, but i did figure it out some minutes ago. xD

Anyone please help me in my submission of E my submission

Your DFS doesn't seem to be working in linear time. For example, consider

Let's say your DFS starts from 1, then visits 3. Then it will backtrack to 1, and visit 2. Then your DFS again visits 3, even though it is already visited. So this can time out for large graphs.

is there any way to fix it? ( because i think we have to see by backtrking is there any good option availiable. so if we use vis array it cant be fixed)

Try not to go to the same state again, if a state is visited use the already computed value for the state. In my case a state holds the max value of all elements from that state to end. You could do something similar

thank's alot for helping me out in this : )

May Codeforces be with uOr just use dijkstra instead of dfs

Dijkstra also requires to record the already visited vertices, so it's essentially the same, I think

I tried my best to solve E, but I can't. Can anyone give me a hand?

Here's my submission

Try to solve with Dijsktra's Algorithm. There are better solutions but using Dijkstra works for this

Only one test-case is failing for D, can anyone figure out the mistake?

https://atcoder.jp/contests/abc188/submissions/19339643

long a=pc*d;

long b=k*d;

This line is causing use WA due to interger overflow. Try to do —

long val = min(pc, k) * d;

Thanks man, I didn't even thought about the max value of pc.

I too got 3 WA for this silly mistake.

Short text editorial. Video livesolve and editorial on my YT channel

C was very easy but , question statement was very confusing , am i only who feels like this or really it was confusing.

It's funny. Cause I first read the statement and sure not able to solve it then try to match the test cases in a rough implementation expecting WA and it works

For E, I tried finding the maximum value in subtree of a node and tried that value with current node value. Why is it wrong? Can someone explain please?

One possible bug is if you include node into its subtree, but Example #3 should rule this out. Also if you use dfs for a subtree max and don't use dp then you get quadratic solutions which TLEs. Other possible bugs include wrong initialization of dp (for example, initializing best answer to 0 instead of -inf or initializing subtree max to something nonzero).

Thanks for reply :) . Sadly I couldn't find anything among them :( . Here is the submission in case needed.

At least two mistakes

You should use min in dfs, not max,

and then you need arr[i] — values[i], not values[i] — arr[i]

Why though? Wouldn't I want maximum selling price for current node? Also, isn't selling price equal to values[i]-arr[i](Selling Price — Cost Price) ?

Nope, you are talking about a different solution with 'reversed' graph.

Well, True, I solved it with the reversed graph and didn't think there was another approach, my bad

If you have edges $$$1 \to 3$$$ and $$$2 \to 3$$$ then $$$3$$$ is in subtree of $$$2$$$ but you won't go to this subtree because $$$3$$$ is already visited after processing $$$1$$$. Lines 16 and 17 should be outside of the if statement. Got AC: https://atcoder.jp/contests/abc188/submissions/19358632

Thanks, it worked T_T

If by sub-tree you mean all nodes reachable from current node , then your logic is correct , there must be coding mistake . Usually sub-tree word we use in case of tree but that was not the input .

Yea. Sorry about that. U r right.I meant nodes reachable from current one.I linked my submission above.

I ignored in E that Xi<Yi, so I tried to implement using bfs (and dfs, too).

The idea is, to find the minimum possible buying price foreach vertex, and collect the max diff of each vertex price minus the minimum price. But that failed for all implementations. What is wrong with that aproach?

Example, dfs: https://atcoder.jp/contests/abc188/submissions/19357783

You need to reverse the edges to calculate values starting from the selling point (largest numbers)

If you do it your way, you can loose some values on the way

say roads are 1->3->5 and 2->3->6

If you start from 1, you will update values for 3 and 5 from 1, values from 2 will never reach 6 because 3 is already visited

The idea is that the value from 3 will reach 5 if it is better than any value reached 5 before.

In the case above, that would be that a[3]<a[1] and a[3]<a[2].

Say a[2] < a[1] < a[3] < a[5] and the roads are as above

And you start your dfs from a[1]

You visit a[1]->a[3]->a[5], then you visit a[6]

How would a[2] reach a[6] if a[3] is already visited?

I allways visit all childs where the minimum of all previus values is bigger that the current value.

This means, for the smallest price I traverse the whole subtree, for bigger prices I do not traverse a subtree a second time, if the first time the min price was smaller than the second time.

Yeah, you're right...

Failed to get this idea initially

Though it should degrade performance of dfs and it will not be linear anymore if you can visit same nodes multiple times

Yes, my last try would most likely TLE because of this. However, some of my previous attempts used a priority_queue, which basically makes that each vertex is traversed at most once.

But in all implementations I considered 0 to be the smallest answer possible.

If i am not wrong , your submission is failing on (input format same as question) :

3 3 2 1 100 1 3 2 3 1 2

Answer is 99 , yours give 98 .

Because in your DFS solution , vertex 3rd is not taken into account when you are at vertex 2nd .

spookywooky , in-fact your solution works on same graph if i reorder the edges in input i.e it's gives 99 on following input :

`3 3 2 1 100 1 2 1 3 2 3`

Ah, ok, found the problem.

"buy 1 kilogram of gold at some town, traverse one or more roads, and sell 1 kilogram of gold at

anothertown."So, like the examples say, the result can be negative. I did not consider that.

Not a very good wording in my opinion btw

If you don't look at all the examples, it is natural to think that if you cannot get positive delta, you just don't buy the gold. They should've emphasized it

Should've looked at the samples :D

I did... after the contest ;)

Sorry for counter-example above , probably i ran solution of comment above yours.

Your solution is giving WA , even after initializing ans = INT_MIN , so what was the final issue ?

I allways considered to sell at the same city as bought, so my min ans is allways 0, never negative.

Maybe in the one or other submission there was other bugs, too, but the main reason for failing was this.

ok,it was mentioned "traverse one or more roads" , thus we can't sell at same city . I also missed that at first reading.

Can anybody give a test case for which my solution is failing. It is failing only on one test case (random_25) Link Edit: Found the mistake (interval * hope[i].second was too big to fit in long long

Solution for D :

Like we do Range update query on array in O(1) Reference We will apply this thing using a map because we can't take array as constraints are big for array size.

My Solution

We increase the m[start[i]] with c[i] and decrease the m[end[i]+1] with c[i] so as if we traverse from the start we can take sum of values in m. We will store the previous day as you can see in my solution. So now our question comes to whether we will take subscription or not for days=it.F-prev.F.

Solution for E :

My solution

For each node we store a high[i] ( it represents the highest a[i] in the cities reachable from i). We can do this using simple DFS. Now for each city we can take current city as the city where we buy our gold and in high[i] we have the city that is reachable from current city, we will sell at the city that has maximum value. We will now take all cities and their child cities and update our answer as ans=max(ans, high[it]-a[curr]).

We bought the gold from city say (i), now we don't sell at the same city, that's why we will be looking at the child of current city for selling.

My E is failing on two test cases. Can someone help me? Submission

`ckmax(maksi[s], a[s]);`

You assume that you can sell at the same city as you bought, but that is not the case, you can only sell in another city.

I think I considered that case. I "split" the graph into two parts such that somewhere in the left part we bought gold and somewhere in the right we sold gold. I split the graph on the edges. (Left part is the left side of the graph when we delete the edge and the right part is the right side of the graph when we delete the edge.)

You maintain two dps, max price per vertex you can get if bought at that vertex, and min price you can have paid if sell at that vertex.

But for both dps you must not put a[i] at position i into mini[i] or maksi[i], because you cannot sell for a[i] if you have bought at i.

Does the third example works with your code?

And btw, you do not need that "split". The max/min diff you find in one of mini or maksi you will also find in the other one. It is enough to check the max selling price or min buying price foreach vertex, no need to check both.

Yeah, the third example works with my code. You got my dp definition wrong. I defined dp (mini) as the minimum value we can get if we "split" the graph at the current node (including and that node). That dp is the left side. The same goes for maksi, but it shows the maximum value. This is the right side. In the end, I overcomplicated my solution, but I am trying to find the mistake. Thank you for trying to help me.

Video Editorial

Great work loved it

For F I understand that if y is even and the optimal answer is going to have a division operation then it is optimal to perform division right now. If y is odd then try both y-1 and y+1. It suggests that after at most 2 operation value will be halved. This way the height of the recursive tree will be log(y)=~60

I tried a few examples and it seems the number of states at each height is not growing very fast because of collision

Is there any tight upper bound for the number of states at each height and overall states in the tree?

Please ignore.. will ask here https://codeforces.com/blog/entry/86642

My Submission

For problem E

I think of it as a DAG

And then dp on top

Why is my solution incorrect

Is there anyone who can help me

You are not updating the mi[] correctly:

I found a counterexample to my accepted code (https://atcoder.jp/contests/abc188/submissions/19346819) for F.

On test case "1 1243", my accepted code prints 16, but the correct answer is 14, as witnessed by the following path: "(((1*2*2+1)*2*2*2-1)*2*2*2-1)*2*2-1"

The issue is that my code does not correctly compute the shortest way to make a number using positive and negative powers of 2; I thought the only issue is that you can replace blocks of ones with 2 operations (a positive and a negative power) but actually the situation is more subtle...you can make 11011011 with only 4 operations (not 6)

Implementation of the official editorial solution for problem E: SUBMISSION (commented as well)

Can someone please explain what is wrong with my code for problem E? I am getting WA on one pretest

Here's the link

https://atcoder.jp/contests/abc188/submissions/19426526

Why is this algorithm not working? Problem — D

The first and/or last intervals are not calculated proper. Consider the first interval starts at day 10. Then you add 10*something to the solution, what is not correct.