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Proof that “local” minimum spanning tree is “global” minimum spanning tree

I’m trying to understand a proof from the book “Graph Theory with Application to Engineering & Computer Science” by Narsingh Deo.

The chapter is about trees in non oriented graphs.

A bit of terminology so that you can understand the theorem and the beginning of the proof from the book:

The author calls minimum spanning trees, shortest spanning trees.

The author calls a branch of a spanning tree any edge of the tree.

A chord of a spanning tree is any edge of the underlying graph that is not in the tree.

A fundamental circuit associated to a spanning tree is a circuit formed by adding one of its chords to a spanning tree (for the author a “circuit” is a closed path, there is no repetition of vertices, it’s what most other sources I’ve read call a cycle). So, a fundamental circuit associated to a spanning tree is a actually a cycle formed by adding one of its chords to a spanning tree.

The distance between two spanning trees $S$ and $T$ of the same graph is (regarding $S$ and $T$ as sets of edges), is $|Ssetminus T|$ (which happens to be equal to $|Tsetminus S|$).

There’s a step in the proof of Theorem 3-16 that I don’t understand.

Theorem 3-16:

A spanning tree T (of a given weighted connected Graph G) is a shortest spanning tree (of G) if and only if there exists no other spanning tree (of G) at a distance of one from T whose weight is smaller than that of T

Proof:

Let $T_1$ be a spanning tree in G satisfying the hypothesis of the theorem (i.e. there is no spanning tree at a distance of one from $T_1$ which is shorter than $T_1$). The proof will be completed by showing that if $T_2$ is a shortest spanning tree (different from $T_1$) in G, the weight of $T_1$ will also be equal to that of $T_2$. Let $T_2$ be a shortest spanning tree in G. Clearly, $T_2$ must also satisfy the hypothesis of the theorem (otherwise there will be a spanning tree shorter than $T_2$ at a distance of one from $T_2$, violating the assumption that $T_2$ is shortest).

Consider an edge $e$ in $T_2$ which is not in $T_1$. Adding $e$ to $T_1$ forms a fundamental circuit with branches in $T_1$. Some, but not all, of the branches in $T_1$ that form the fundamental circuit with $e$ may also be in $T_2$; each of these branches in $T_1$ has a weight smaller than or equal to that of $e$, because of the assumption on $T_1$. Amongst all those edges in this circuit which are not in $T_2$ at least one, say $b_j$, must form some fundamental circuit (with respect to $T_2$) containing $e$.

I’m stuck at the last sentence I just quoted:

“Amongst all those edges in this circuit which are not in $T_2$ at least one, say $b_j$, must form some fundamental circuit (with respect to $T_2$) containing $e$.”

I don’t see why among those cycles, there should necessarily be one that contains $e$. Why is that?

Potential problems with global scope via isolated file bundling in .NET for JavaScript files?

If anyone can suggest a much better worded question, but what I am trying to ask here is, I turned in a well designed JavaScript frontend, but the .NET developer could not get it to bundle using the Mads Kristensen bundler tool.

The bundler complained about the import statements. There was nothing wrong with the import statements they were used correctly and made for a working frontend. The error kept referring to an unexpected end of input citing the semicolons of the import statements.

So the individual decided to remove all import statements and bundle all the files in isolation, which I believe runs into the issue as answered by Martijn:

why are globals bad in javascript

If I am correct about the above applying to this case, could someone provide a concrete example as to what are some potentials or cite a use case where doing things this way went really bad? Saying that everything has access to everything and is contrary to encapsulation goes over the head of non-technical audience.

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App Windows – Global Mapper 22.0.1 | NulledTeam UnderGround

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How to increase a number of daily links and LPM with global site lists?

Hello,
i have a question.
Is there any way to increase a number of daily links that GSA SER creates from global site lists? I use lists from https://www.serpowerlists.com/ and they are great. But usually i have only about 200-300 verified links per day and LPM is 0,81. I do not set any limits on daily links in settings:
https://prnt.sc/uz9uon
All filtering options are unchecked:
https://prnt.sc/uz9w8p
I connected dropbox folders to GSA SER:
Identified – Contextual_URLs
Submitted – Top_Tier_URLs
Failed – Verified_URLs
https://prnt.sc/uz9y63
Checked Identified, Submitted and Failed and Use URLs from global site lists if enabled:
https://prnt.sc/uz96tm
I run GSA SER on Windows VPS with 4 core and 8 GB RAM at 40 threads with rotating dedicated proxies from https://stormproxies.com/rotating_reverse_proxies.html (40 threads proxy package), 1 catcha-ll e-mail from catchallboxes.com and Xevil for captcha recognition.
For some reason GSA SER runs mostly at 7-10 threads instead of 40 in settings:
https://prnt.sc/uz99pq
CPU and RAM are not higher than 50%.
I decided to check how it will change the situation and directly uploaded target URLs into the project (Import target URLs – From site lists – Identified, Submitted and Failed). GSA speeds up to 40 threads, LPM is still low:
https://prnt.sc/uz9ejj
In total its a big number of targeted URLs (713K URLs), but i still have about 200-300 daily links from uploading targeted URLs.
Sometimes i see message “No targets to post to (no search engines chosen, no url extraction chosen, no scheduled posting)” in my projects:
https://prnt.sc/uz9q76
At the same time dropbox folders with site lists are updating every few hours.
If i click “Show URls – Show remaining target URLs”, it usually shows 0 URLs, or 8-10 URLs.
Is it possible to solve this problem? Am i doing something wrong?
Thank you!

python – Can’t seem to find a way not to use global in my code

This is a riot analysis app. Collect data using the riot API and shows the user various data. Nothing too in depth.

gui file

import tkinter as tk
from tkinter import font as tkfont
from getId import id_collected
from games import Game
from wins import is_player_good


class RiotApp(tk.Tk):

    def __init__(self, *args, **kwargs):
        tk.Tk.__init__(self, *args, **kwargs)
        self.title_font = tkfont.Font(family='Helvetica', size=18, weight="bold", slant="italic")
        container = tk.Frame(self)
        container.pack(side="top", fill="both", expand=True)
        container.grid_rowconfigure(0, weight=1)
        container.grid_columnconfigure(0, weight=1)
        self.frames = {}
        for F in (StartPage, MenuPage, KillPage, DeathPage, CsPage, HonestPage):
            page_name = F.__name__
            frame = F(parent=container, controller=self)
            self.frames(page_name) = frame
            frame.grid(row=0, column=0, sticky="nsew")

        self.show_frame("StartPage")

    def show_frame(self, page_name):
        frame = self.frames(page_name)
        frame.tkraise()

class StartPage(tk.Frame):

    def __init__(self, parent, controller):
    
        tk.Frame.__init__(self, parent)
        c = DataCollected()
        self.controller = controller
        self.label = tk.Label(self, text="Enter summoner name:", width = 20, font = ("bold", 20))
        self.label.place(x=90,y=53)
        self.entry = tk.Entry(self)
        self.entry.place(x=190,y=130)
        
        self.button = tk.Button(self, text="Search",width = 20, bg = 'brown', fg = 'white',
                            command=lambda: data_collected(self,controller))
        self.button.place(x=180,y=200)

        def data_collected(self,controller):
        
            name = self.entry.get()
            Key = 'RGAPI-83fcae12-8840-49df-b539-a47f535e1c4e'
            a = id_collected(name, Key)
            if a != 'NO':
                controller.show_frame("MenuPage")
                c.collect_data(name, Key)
            else:
                controller.show_frame('StartPage')

class DataCollected():
    
    def collect_data(self, name, Key):

        num_games = 20
        game = Game()
        accId = id_collected(name, Key)
        game_list = game.find_game_ids(accId, Key, num_games)
        global stat_list
        stat_list = game.game_data(game_list, Key, name, num_games)
        global honest
        honest = is_player_good(stat_list(5))

class MenuPage(tk.Frame,DataCollected):

    def __init__(self, parent, controller):
        tk.Frame.__init__(self, parent)
        self.controller = controller
        label = tk.Label(self, text="Main Menu", font=controller.title_font)
        label.place(x=180,y=50)
        button = tk.Button(self, text="Kill Average",width = 20, bg = 'brown', fg = 'white',
                           command=lambda: controller.show_frame("KillPage")).place(x=180,y=100)
        button = tk.Button(self, text="Death Average",width = 20, bg = 'brown', fg = 'white',
                           command=lambda: controller.show_frame("DeathPage")).place(x=180,y=150)
        button = tk.Button(self, text="Cs Average",width = 20, bg = 'brown', fg = 'white',
                           command=lambda: controller.show_frame("CsPage")).place(x=180,y=200)
        button = tk.Button(self, text="Honest Truth",width = 20, bg = 'brown', fg = 'white',
                           command=lambda: controller.show_frame("HonestPage")).place(x=180,y=250)
        button = tk.Button(self, text="Back",width = 20, bg = 'brown', fg = 'white',
                           command=lambda: controller.show_frame("StartPage")).place(x=180,y=300)

class KillPage(tk.Frame, DataCollected):
    
    def __init__(self, parent, controller):
        tk.Frame.__init__(self, parent)
        self.controller = controller
        self.label = tk.Label(self, text = 'Kills Average', width=20,font=("bold", 20))
        self.label.place(x=90, y=100)
        self.label1 = tk.Label(self, text = ' ', width=20,font=("bold", 20))
        self.label1.place(x=90, y=150)
        self.label1.after(1000, self.refresh_label)
        self.button = tk.Button(self, text = "Back", width = 20, bg = 'brown', fg = 'white',
                        command=lambda: controller.show_frame("MenuPage")).place(x=180,y=300)

    def refresh_label(self):
        self.label1.configure(text = stat_list(1))
        self.label1.after(1000,self.refresh_label)

class DeathPage(tk.Frame):

    def __init__(self, parent, controller):
        tk.Frame.__init__(self, parent)
        self.controller = controller
        self.label = tk.Label(self, text = 'Deaths Average', width=20,font=("bold", 20))
        self.label.place(x=90, y=100)
        self.label2 = tk.Label(self, text="", width=20,font=("bold", 20))
        self.label2.place(x=90, y=150)
        self.label2.after(1000, self.refresh_label)
        self.button = tk.Button(self, text="Back", width = 20, bg = 'brown', fg = 'white',
                           command=lambda: controller.show_frame("MenuPage")).place(x=180,y=300)

    def refresh_label(self):
        self.label2.configure(text = stat_list(0))
        self.label2.after(1000,self.refresh_label)

class CsPage(tk.Frame):

    def __init__(self, parent, controller):
        tk.Frame.__init__(self, parent)
        self.controller = controller
        self.label = tk.Label(self, text = 'Cs Average', width=20,font=("bold", 20))
        self.label.place(x=90, y=100)
        self.label3 = tk.Label(self, text="", width=20,font=("bold", 20))
        self.label3.place(x=90,y=150)
        self.label3.after(1000, self.refresh_label)
        self.button = tk.Button(self, text="Back", width = 20, bg = 'brown', fg = 'white',
                           command=lambda: controller.show_frame("MenuPage")).place(x=180,y=300)

    def refresh_label(self):
        self.label3.configure(text = stat_list(4))
        self.label3.after(1000,self.refresh_label)

class HonestPage(tk.Frame):

    def __init__(self, parent, controller):
        tk.Frame.__init__(self, parent)
        self.controller = controller
        self.label = tk.Label(self, text = 'Honest Truth', width=20,font=("bold", 20))
        self.label.place(x=90, y=100)
        self.label4 = tk.Label(self, text = " ", width=20,font=("bold", 20))
        self.label4.place(x=90,y=150)
        self.label4.after(1000, self.refresh_label())
        self.button = tk.Button(self, text = "Back", width = 20, bg = 'brown', fg = 'white',
                           command=lambda: controller.show_frame("MenuPage")).place(x=180,y=300)

    def refresh_label(self):
        self.label4.configure(text = honest)
        self.label4.after(1000,self.refresh_label)

if __name__ == "__main__":

    stat_list = (1,1,1,1,1,1,1)
    honest = ' '
    root = RiotApp()
    root.geometry("500x500")
    root.mainloop()

game file

import requests

class Game:

    def find_game_ids(self, accId, key, num_games):
        i = 0
        GAMEID = ()
        num_games = 20
        url_match_list = ('https://na1.api.riotgames.com/lol/match/v4/matchlists/by-account/' + (accId) + '?queue=420&endIndex=20&api_key=' + (key))
        response2 = requests.get(url_match_list)
        # Adding 20 games into the list
        while num_games > 0:
            GAMEID.append('https://na1.api.riotgames.com/lol/match/v4/matches/'+str(response2.json()('matches')(i)('gameId')) + '?api_key=' + (key))
            i = i + 1
            num_games = num_games - 1

        return GAMEID

    def game_data(self, game_list, key, sumName, num_games):

        wins = ()
        deaths = ()
        deaths = ()
        kills = ()
        assists = ()
        visions = ()
        csTotal = ()
        # Finding the data of said summoner in each game id
        for urls in game_list:
            response = requests.get(urls)
            resp_json = response.json()
            Loop = 0
            index = 0
            while Loop <= 10:

                if resp_json('participantIdentities')(index)('player')('summonerName') != sumName:
                    Loop = Loop+1
                    index = index+1
                elif resp_json('participantIdentities')(index)('player')('summonerName') == sumName:

                    deaths.append(resp_json('participants')(index)('stats')('deaths'))
                    kills.append(resp_json('participants')(index)('stats')('kills'))
                    assists.append(resp_json('participants')(index)('stats')('assists'))
                    visions.append(resp_json('participants')(index)('stats')('visionScore'))
                    csTotal.append(resp_json('participants')(index)('stats')('totalMinionsKilled'))
                    wins.append(resp_json('participants')(index)('stats')('win'))

                    break
        # Finding avg of each stat
        deaths = sum(deaths)/num_games
        kills = sum(kills)/num_games
        assists = sum(assists)/num_games
        visions = sum(visions)/num_games
        csTotal = sum(csTotal)/num_games
        wins = sum(wins)/num_games
        stat_list = ()
        stat_list.append(deaths) #0
        stat_list.append(kills) #1
        stat_list.append(assists) #2
        stat_list.append(visions) #3
        stat_list.append(csTotal) #4
        stat_list.append(wins) #5

        return stat_list

get id file

import requests

def id_collected(sumName, key):
    # COLLECTING DATA TO BE INSERTING FOR MATCHLIST DATABASE
    url = ('https://na1.api.riotgames.com/lol/summoner/v4/summoners/by-name/'+(sumName)+'?api_key='+ 
(key))
    response = requests.get(url)
    if response.status_code == 200:
        accId = (response.json()('accountId'))
        return accId
    else: 
        accId = 'NO'
        return accId

wins file

import random


def is_player_good(winlist):

    if winlist < 0.33:
        message = ('DIS MANE STINKS', 'run while you can', 'I repeat, YOU ARE NOT WINNING THIS', 'I predict a fat L', 'Have fun trying to carry this person', 'He is a walking trash can', 'He needs to find a new game', 'BAD LUCK!!!')
        return (random.choice(message))

    elif winlist > 0.33 and winlist <= 0.5:
        message = ('Losing a bit', 'Not very good', 'He needs lots of help', 'Your back might hurt a little', 'Does not win much')
        return (random.choice(message))

    elif winlist > 0.5 and winlist <= 0.65:
        message = ('He is ight', 'He can win a lil', 'You guys have a decent chance to win', 'Serviceable', 'Should be a dub')
        return (random.choice(message))

    elif winlist > 0.65:
        message = ('DUB!', 'You getting carried', 'His back gonna hurt a bit', 'winner winner chicken dinner', 'Dude wins TOO MUCH', 'You aint even gotta try', 'GODLIKE')
        return (random.choice(message))

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  • Data Centre Choice
  • 24*7 Support
  • Power & Scalability
  • Free SSL Certificate

Our Support Team is prepared to help you in any circumstance and anytime to tackle any issue in the best possible manner. We additionally furnish Instant Talk administration to interact with you, virtually.

For Immediate Assistance or information, contact us: info@hostitsmart.com or visit our website for live chat.