Lecture 15: Single-Source Shortest Paths Problem

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  • เผยแพร่เมื่อ 5 พ.ย. 2024

ความคิดเห็น • 106

  • @SparshPaliwal
    @SparshPaliwal 7 ปีที่แล้ว +118

    Everyone please stop saying the lecture is worthless. This lecture is a Primer for shortest path algorithms. Understanding negative weights, negative loops was pretty important. Also learning about all different problems like exponential number of vertices. Surely if you had gone through the topic already might be useless to you, so you can skip it. But if you are going through shortest path algos for the first time, its a good lecture.

    • @ryanc7840
      @ryanc7840 2 ปีที่แล้ว +4

      Right. Without this one, subsequent lectures are hard to grasp at least regarding the notations. I concur that this one is a good and necessary primer.

  • @zvezdalion
    @zvezdalion 9 ปีที่แล้ว +80

    This professor's lecture is so structured! He really stresses the important things. So easy to get the big picture, but also gives you the often-overlooked details (e.g., it's not the neg. weight edges, but the *cycles* that are a problem; Dijkstra is essentially *linear*)

  • @sergeykholkhunov1888
    @sergeykholkhunov1888 3 ปีที่แล้ว +31

    02:30 motivation, Dijkstra, Bellman-Ford
    08:25 definitions (path, weight of path, shortest-path problem)
    11:10 about time complexity
    14:21 weighted graphs
    17:56 example
    23:55 predecessor relationship definition
    26:40 negative weights
    32:00 negative weight cycles
    37:30 general structure of shortest-path algorithms, relaxation
    45:02 example (problems with exponential number of paths)
    50:35 optimal substructure (shortest path's subpaths, triangle inequality)

  • @md.sayeedrahman2553
    @md.sayeedrahman2553 4 ปีที่แล้ว +8

    loved it! personally i feel attached to this professor! his lecture is so amazing!!!

  • @Qladstone
    @Qladstone 7 ปีที่แล้ว +6

    If each edge represents a multiplier, and you want to minimize the product of the multipliers along a path; that is equivalent to minimizing the sum of the logarithms of those multipliers. Logarithms have a range from -inf to +inf, and so include both negative, zero, and positive values.
    One commonly cited example is where the multipliers represent currency conversion rates; where each vertex represents one currency, and edges represent conversion rates between currencies.

    • @pengliu9987
      @pengliu9987 ปีที่แล้ว

      Great example of negative weight graph. Thanks for sharing.

  • @marvinmurphy4330
    @marvinmurphy4330 10 ปีที่แล้ว +16

    neg weights? 28:55 - What about graphs being represented as electric circuits? Some branches have passive components like resistors that cause a potential drop and some branches have active components like batteries that cause a potential gain. Intersections or splits in the circuit are vertices. The connections between splits are edges and the weights are the voltage gain (+ve) or drop (-ve) of that connection.

    • @surjayanghosh5872
      @surjayanghosh5872 4 ปีที่แล้ว +1

      thats an interesting obeservation! ..... but how would we use the different paths/ because in electrical circuits, current (I) is gonna flow in all the branches, i mean we cant tell it to use only a certain path. So, can you think of any way it could be used? but good analogy nonetheless.

    • @iamparitosh
      @iamparitosh 3 ปีที่แล้ว

      Also the total potential gain of any cycle is zero here

  • @SzechSauce
    @SzechSauce 5 ปีที่แล้ว +6

    Isn’t there a mistake at 48:20? How can you relax the edge from v4 to v5 to then have d(v5) be 12? That would mean v4 can be reached in 12, and v5 can also be reached in 12? That can’t be correct...?
    Really great lecture otherwise, quite surprised at all the negative comments on here...

  • @anunaybagga2948
    @anunaybagga2948 7 ปีที่แล้ว +42

    i understood each and everything sir said and hence i don't get the hate comments out here.

    • @mohitsingh-nf5vl
      @mohitsingh-nf5vl 4 ปีที่แล้ว

      Can u explain me time complexity plz??

    • @zeronothinghere9334
      @zeronothinghere9334 4 ปีที่แล้ว +1

      @@mohitsingh-nf5vl Look at the first video in the playlist. This is the 15th video in the series after all.

  • @user-td8zb9xo9k
    @user-td8zb9xo9k 2 ปีที่แล้ว +1

    happy to see indian professor here

  • @loganwishartcraig
    @loganwishartcraig 6 ปีที่แล้ว +17

    This lecture was definitely fine.

  • @Tintak_hatpin
    @Tintak_hatpin 4 ปีที่แล้ว +4

    I like the way he says "you could imagine"

  • @cskwillreturn2473
    @cskwillreturn2473 7 ปีที่แล้ว +18

    At 12:43, did he say O(E^2) possible weights? I think it should be V^2.

    • @moongumoongu
      @moongumoongu 7 ปีที่แล้ว +4

      yeah, jotted that down and got really confused later. if there are |E| edges, the number of possible unique weights is |E|... and E=O(V^2), so yeah, i think he meant to say roughly V^2 possible weights.

    • @bigdildo
      @bigdildo 5 ปีที่แล้ว +1

      agreee

    • @csl1384
      @csl1384 5 ปีที่แล้ว +1

      Glad to see I am not the only one who thought this.

    • @ShivamPatel-ww1up
      @ShivamPatel-ww1up 4 ปีที่แล้ว

      my doubt exactly

  • @aerronpro1133
    @aerronpro1133 2 ปีที่แล้ว

    Wow this lec is genius, generalises whole shortest path in a lec

  • @lockersrandom6161
    @lockersrandom6161 4 ปีที่แล้ว +2

    Thank YOU MIT.

  • @MrKingoverall
    @MrKingoverall 4 ปีที่แล้ว

    Operations Research Major here: LOVE YOU MAN !! :)

  • @eepassion9720
    @eepassion9720 5 ปีที่แล้ว +2

    29:36 you will see Victor giving a example about negative weight. We would know more about MIT TA work. That is so different in my college. I don't even know who is my TA in the entire of my college.

  • @yashjain6945
    @yashjain6945 6 ปีที่แล้ว +3

    What does he mean by at least gets credit for it 4:27

  • @josh7910
    @josh7910 10 ปีที่แล้ว +12

    48:20
    I think he got the 13 ->12 from v5 wrong

    • @fabischn90
      @fabischn90 10 ปีที่แล้ว +1

      It's perfectly correct. After relaxing the edge (v4, v6) he relaxed edge (v5, v6)

    • @PK-en1bm
      @PK-en1bm 10 ปีที่แล้ว

      ***** I am thinking about relaxation of the edge (v4, v5)??

    • @fabischn90
      @fabischn90 10 ปีที่แล้ว +2

      Prasanjit Khuntia You would understand, if you just had continued watching for another 2 minutes. Keep calm :)

    • @JustinLiang0
      @JustinLiang0 10 ปีที่แล้ว +2

      Yeah odd... I am not sure how he can "relax" the edge like that because nothing seems to satisfy d[v] > d[u] + w(u,v) at that point.

    • @botelhorui
      @botelhorui 9 ปีที่แล้ว +2

      Justin Liang Yes, its a mistake

  • @neuron8186
    @neuron8186 3 ปีที่แล้ว +6

    i don't understand the hate both teacher are best and you haters are nothing more than a keyboard warrior how did you get the audacity to hate people providing free education

  • @isbestlizard
    @isbestlizard 4 ปีที่แล้ว

    why are cycles even a thing? if they add positive path length you'd obviously never take them, if they add negative path length you'd obviously go around around them forever so surely it makes no sense to even have a graph with cycles when asking the question 'what's the shortest path'.. without adding extra criteria like 'without retracing the same edge' or 'without visiting the same node more than x times'

  • @RalyJoey
    @RalyJoey 8 ปีที่แล้ว +3

    48:22 how did the instructor obtain 12 for the second last vertex of this directed graph?

    • @fredwu6812
      @fredwu6812 8 ปีที่แล้ว +1

      that was a mistake.

    • @anunaybagga2948
      @anunaybagga2948 7 ปีที่แล้ว

      Fred Wu no man it is not a mistake watch it again

    • @tu-ningting2867
      @tu-ningting2867 4 ปีที่แล้ว +4

      Anunay Bagga It's been 2 years. Hope you now see it is a mistake. If not, watch it again

  • @ChristopherPelnar
    @ChristopherPelnar 2 ปีที่แล้ว

    Negative weights are like hitting wormholes in space-time.

  • @SanghoBose5
    @SanghoBose5 6 ปีที่แล้ว +3

    Travelling upstream in river be considered as negative weight path.

  • @hoboishjoeishme
    @hoboishjoeishme 10 ปีที่แล้ว +15

    45:23 sounds like someone's squeezing out a little fart

    • @FactVault5584
      @FactVault5584 7 ปีที่แล้ว +1

      lol

    • @swappybird
      @swappybird 7 ปีที่แล้ว +3

      I came for the this exact comment.

    • @mo_pak_tech
      @mo_pak_tech 5 ปีที่แล้ว +2

      legend

    • @fredwu6812
      @fredwu6812 5 ปีที่แล้ว +2

      What are you here for? Listening to farting or lectures?

  • @Mrimadeyoufall
    @Mrimadeyoufall 2 ปีที่แล้ว

    So helpful

  • @이택영-l9h
    @이택영-l9h 2 ปีที่แล้ว

    His lecture is very meticulous

  • @gouravgoel2974
    @gouravgoel2974 4 ปีที่แล้ว

    this is gold

  • @rithwikteja2168
    @rithwikteja2168 3 ปีที่แล้ว

    I need to find victor now

  • @yettodecideahandle
    @yettodecideahandle 4 ปีที่แล้ว +1

    Srini is very good and seem to explain thing simple, Eric makes the same lectures complicated.

    • @elliott8175
      @elliott8175 4 ปีที่แล้ว +4

      Erik is the boss. I love his intuitive explanations. He gets straight to the ideas in a really down-to-earth way: "Dynamic programming is really just careful brute force". If you get DP, you realise how other teachers make it out to be much more complicated than it is.

  • @brendawilliams8062
    @brendawilliams8062 3 ปีที่แล้ว

    Negative takes so much work with squares and cubes. People don’t want what appears to be infinities of work. They want a genesis of general and fast tracked work.

  • @mayonesacosmica
    @mayonesacosmica 10 ปีที่แล้ว +1

    how about circuits where there are multiple paths w/ positive or negative voltage differentials for a motivation? if so, I can send you my address for a cushion... ;)

    • @Joshuaposada
      @Joshuaposada 9 ปีที่แล้ว

      Wouldn't you just use the magnitude? I'm not really sure, but that's something to think about

  • @yangyinjedi
    @yangyinjedi 9 ปีที่แล้ว +5

    I like the other professor a little better I feel like his lectures are more random and less structured...I felt like this professor was just teaching by the book...But they're are both great sup lectures

  • @nikhilrajput5030
    @nikhilrajput5030 4 ปีที่แล้ว +1

    Can anyone help me why E=O(V^2)??

    • @abhaikollara5811
      @abhaikollara5811 4 ปีที่แล้ว +3

      Number of possible edges between n vertices is nC2 i.e the number of ways to pick 2 vertices from a given set of n vertices. Unless I screwed up, replacing r=2 in the equation for nC2 should derive into (V^2 - 1)/2 which is O(V^2).

    • @nikhilrajput5030
      @nikhilrajput5030 4 ปีที่แล้ว +2

      @@abhaikollara5811 Thanks man!

    • @legendarylea560
      @legendarylea560 4 ปีที่แล้ว

      @@abhaikollara5811 I think it's not about nC2 (combinations) but nP2 (permutations), so nPr -> n^r -> n^2. Am I wrong?

    • @cvprasanthkumar
      @cvprasanthkumar 4 ปีที่แล้ว +3

      I think it is because every Vertex can have atmost V Edges(Unless we are talking about Multi Graph). So you have V Vertex with each having V Edges, so overall edges are V^2

  • @davidporterrealestate
    @davidporterrealestate 7 ปีที่แล้ว

    41:20 49:00

  • @isbestlizard
    @isbestlizard 4 ปีที่แล้ว +1

    why would anyone imagine the weight defined for an edge WOULD affect the calculation speed of an algorithm? that'd be like thinking the 'weight' of an object would affect the time it took to sort it. 'this is a really heavy object so it takes longer to add it to the list' wot no 'this is a really long path so it takes longer to calculate its length' WOT NO WHY WOULD YOU THINK THAT :V

    • @adhishmalviya2408
      @adhishmalviya2408 4 ปีที่แล้ว

      49:30 the random relaxing of edges can be exponential and dependent on order of weights

  • @csl1384
    @csl1384 5 ปีที่แล้ว

    14:05 If Dijkstra is "basically" linear time [O(VlogV + E)], then Bellman-Ford [O(VE)] should be "basically" quadratic time, not cubic time.

    • @philodev874
      @philodev874 4 ปีที่แล้ว +6

      No, the professor explains how E’s asymptotic upper bound is V^2(imagine each vertex containing an edge to every other vertex) therefore V * E = V * V^2 = V^3 (asymptotic upper bound)

    • @csl1384
      @csl1384 4 ปีที่แล้ว +1

      @@philodev874 agreed, thanks

    • @philodev874
      @philodev874 4 ปีที่แล้ว

      c sl no problem!

  • @rohangupta3486
    @rohangupta3486 9 ปีที่แล้ว +2

    how he wrote
    E=O(V^2) ?????????????????

    • @cskwillreturn2473
      @cskwillreturn2473 7 ปีที่แล้ว +4

      Assume two vertices v1 and v2. The total number of possible edges would be 4 which is 2^2 ( 2= total number of vertices). All possible edges v1 -> v2, v2 -> v1, v1->v1, v2->v2. They all add up to 4. But this is only correct if the graph is directed. If undirected, I think it should be v(v-1)/2.

    • @kp2718
      @kp2718 6 ปีที่แล้ว +4

      imagine an adjacency matrix

    • @legendarylea560
      @legendarylea560 4 ปีที่แล้ว

      @IndiaRockLovers I think it's not about nC2 (combinations) but nP2 (permutations), so nPr -> n^r -> n^2. Am I wrong?

    • @legendarylea560
      @legendarylea560 4 ปีที่แล้ว

      @IndiaRockLovers Perfect, thx

    • @mystmuffin3600
      @mystmuffin3600 4 ปีที่แล้ว

      it's about how many (u,v) pairs can be formed. Directed edges would yield nP2 edges given the graph is simple (u cannot be equal to v) and nC2 edges for undirected as (u,v) and (v,u) would only be counted once

  • @ShabbirAhmed-ug8sv
    @ShabbirAhmed-ug8sv 9 ปีที่แล้ว +1

    22:25

  • @abdelrhmansamir1426
    @abdelrhmansamir1426 5 ปีที่แล้ว +1

    I know it's too late i guess. but can anyone explain why does the asymptotic relation between edges and vertices is v^2? don't know if i said it correctly as well just didn't get that point.

    • @vamshidharreddy2580
      @vamshidharreddy2580 5 ปีที่แล้ว +1

      maximum number of edges possible is of the order v^2. which is same as maximum number of handshakes among 'v' people.

    • @shubhankitsingh5196
      @shubhankitsingh5196 5 ปีที่แล้ว +3

      nC2

    • @thehvhnk
      @thehvhnk 3 ปีที่แล้ว +1

      discrete math? Imagine you pick two random nodes in a bunch of nodes so when total of nodes is N therefore total of number of ways is NC2 ~ n^2

    • @Aryan-xj1qz
      @Aryan-xj1qz 3 ปีที่แล้ว +1

      In a complete graph, the number of edges is V choose 2 (The total number of ways by which we can pair two vertices), which equals V*(V-1)/2. This is of order V^2. Hence, E = O(V^2)

  • @jshellenberger7876
    @jshellenberger7876 10 หลายเดือนก่อน

    #POW

  • @douzigege
    @douzigege 6 ปีที่แล้ว +1

    I can feel the pain. The reason why sometimes he is not delivering the lectures is probably the fact that Dr. Devadas is not a mathematician like Erik.

  • @tchappyha4034
    @tchappyha4034 4 ปีที่แล้ว

    11:55 very dangerous!

  • @connectication
    @connectication 9 ปีที่แล้ว +5

    30:32, sooo awkward

  • @ravitejapothala7114
    @ravitejapothala7114 5 ปีที่แล้ว

    Mm,kumms

  • @rj-nj3uk
    @rj-nj3uk 5 ปีที่แล้ว +1

    He did not taught dijkstra properly.

  • @ChrisLeeX
    @ChrisLeeX 8 ปีที่แล้ว +6

    While I'm a big fan of this series and I like the lecturer, this particular lecture is terrible. Just skip it.

    • @roylee3196
      @roylee3196 8 ปีที่แล้ว

      +Chris Lee Yes you're right.

    • @satadhi
      @satadhi 7 ปีที่แล้ว +4

      Hi can you be more specific please ? cause i don't see anything wrong in this , maybe i am missing something ?

    • @xinli6243
      @xinli6243 5 ปีที่แล้ว

      I like Eric.

  • @kp2718
    @kp2718 6 ปีที่แล้ว +1

    You get to appreciate the quality of a lecture after watching something like this. Where's Eric??? I think this guy forgot whats' the point of lectures, so he just rambles.

  • @piyushsharma6811
    @piyushsharma6811 5 ปีที่แล้ว +1

    Disappointed.

  • @chenzhiye633
    @chenzhiye633 7 ปีที่แล้ว +1

    waste of time=,=

  • @jsridhar72
    @jsridhar72 7 ปีที่แล้ว

    Just unberable. Wasted precious time. You wont gain anything by watching this. Just skip it.