Shortest length of wire required to connect points
$begingroup$
There are 2 sets , $A={a_1,a_2,a_3..... ,a_n} $ and $B={b_1,b_2,b_3.. b_n}$ . All elements of these sets are real numbers. Each point in set $A$ need to be connected to one and only one point in set $B$ . How must we connect to use shortest length of wire??
I know the answer is take the smallest element in $A$ and connect to smallest element in $B$ , then do so for second-smallest ones and so on.. I was trying to proof it . I tried using induction.. I also tried trying to decompose other possible connections to the one that is the solution.. but I failed.. miserably.
sequences-and-series fixedpoints
asked Dec 14 '18 at 7:03
Kavita JunejaKavita Juneja
1218
$endgroup$
add a comment |
$begingroup$
There are 2 sets , $A={a_1,a_2,a_3..... ,a_n} $ and $B={b_1,b_2,b_3.. b_n}$ . All elements of these sets are real numbers. Each point in set $A$ need to be connected to one and only one point in set $B$ . How must we connect to use shortest length of wire??
I know the answer is take the smallest element in $A$ and connect to smallest element in $B$ , then do so for second-smallest ones and so on.. I was trying to proof it . I tried using induction.. I also tried trying to decompose other possible connections to the one that is the solution.. but I failed.. miserably.
sequences-and-series fixedpoints
asked Dec 14 '18 at 7:03
Kavita JunejaKavita Juneja
1218
$endgroup$
1
$begingroup$
How do you connect two real numbers "via lengths of wire?" Are you asking for some permutation $piin S_n$ such that the summation $sumlimits_{i=1}^n |a_i - b_{pi(i)}|$ is minimized?
$endgroup$
– JMoravitz
Dec 14 '18 at 7:07
$begingroup$
@ShubhamJohri: Notice that the sets have the same number of elements.
$endgroup$
– David G. Stork
Dec 14 '18 at 7:56
$begingroup$
@JMoravitz yes, exactly.. you are right. :)
$endgroup$
– Kavita Juneja
Dec 14 '18 at 10:48
1
$begingroup$
I'll assume $a_1$ and $b_1$ are the smallest elements of $A$ and $B$. If you have some wiring solution where they are not connected, then $a_1$ connects to $b_u$ and $a_v$ connects to $b_1$ for some $u$ and $v$. Show that if you swap the connections to $a_1-b_1$ and $a_v-b_u$ then the total length of wire cannot increase. So to get an optimal solution, you can assume without loss of generality that $a_1$ and $b_1$ are connected. Then induction takes care the rest, showing that the proposed solution is optimal. It is not always the only optimal solution, but is the only general solution.
$endgroup$
– Jaap Scherphuis
Dec 14 '18 at 14:47
$begingroup$
@JaapScherphuis thank you very mcuh
$endgroup$
– Kavita Juneja
Dec 14 '18 at 15:25
add a comment |
$begingroup$
There are 2 sets , $A={a_1,a_2,a_3..... ,a_n} $ and $B={b_1,b_2,b_3.. b_n}$ . All elements of these sets are real numbers. Each point in set $A$ need to be connected to one and only one point in set $B$ . How must we connect to use shortest length of wire??
I know the answer is take the smallest element in $A$ and connect to smallest element in $B$ , then do so for second-smallest ones and so on.. I was trying to proof it . I tried using induction.. I also tried trying to decompose other possible connections to the one that is the solution.. but I failed.. miserably.
sequences-and-series fixedpoints
asked Dec 14 '18 at 7:03
Kavita JunejaKavita Juneja
1218
$endgroup$
There are 2 sets , $A={a_1,a_2,a_3..... ,a_n} $ and $B={b_1,b_2,b_3.. b_n}$ . All elements of these sets are real numbers. Each point in set $A$ need to be connected to one and only one point in set $B$ . How must we connect to use shortest length of wire??
I know the answer is take the smallest element in $A$ and connect to smallest element in $B$ , then do so for second-smallest ones and so on.. I was trying to proof it . I tried using induction.. I also tried trying to decompose other possible connections to the one that is the solution.. but I failed.. miserably.
sequences-and-series fixedpoints
sequences-and-series fixedpoints
asked Dec 14 '18 at 7:03
Kavita JunejaKavita Juneja
1218
asked Dec 14 '18 at 7:03
Kavita JunejaKavita Juneja
1218
edited Dec 14 '18 at 14:14
Kavita Juneja
asked Dec 14 '18 at 7:03
Kavita JunejaKavita Juneja
1218
asked Dec 14 '18 at 7:03
Kavita JunejaKavita Juneja
1218
asked Dec 14 '18 at 7:03
Kavita JunejaKavita Juneja
1218
1218
1
$begingroup$
How do you connect two real numbers "via lengths of wire?" Are you asking for some permutation $piin S_n$ such that the summation $sumlimits_{i=1}^n |a_i - b_{pi(i)}|$ is minimized?
$endgroup$
– JMoravitz
Dec 14 '18 at 7:07
$begingroup$
@ShubhamJohri: Notice that the sets have the same number of elements.
$endgroup$
– David G. Stork
Dec 14 '18 at 7:56
$begingroup$
@JMoravitz yes, exactly.. you are right. :)
$endgroup$
– Kavita Juneja
Dec 14 '18 at 10:48
1
$begingroup$
I'll assume $a_1$ and $b_1$ are the smallest elements of $A$ and $B$. If you have some wiring solution where they are not connected, then $a_1$ connects to $b_u$ and $a_v$ connects to $b_1$ for some $u$ and $v$. Show that if you swap the connections to $a_1-b_1$ and $a_v-b_u$ then the total length of wire cannot increase. So to get an optimal solution, you can assume without loss of generality that $a_1$ and $b_1$ are connected. Then induction takes care the rest, showing that the proposed solution is optimal. It is not always the only optimal solution, but is the only general solution.
$endgroup$
– Jaap Scherphuis
Dec 14 '18 at 14:47
$begingroup$
@JaapScherphuis thank you very mcuh
$endgroup$
– Kavita Juneja
Dec 14 '18 at 15:25
add a comment |
1
$begingroup$
How do you connect two real numbers "via lengths of wire?" Are you asking for some permutation $piin S_n$ such that the summation $sumlimits_{i=1}^n |a_i - b_{pi(i)}|$ is minimized?
$endgroup$
– JMoravitz
Dec 14 '18 at 7:07
$begingroup$
@ShubhamJohri: Notice that the sets have the same number of elements.
$endgroup$
– David G. Stork
Dec 14 '18 at 7:56
$begingroup$
@JMoravitz yes, exactly.. you are right. :)
$endgroup$
– Kavita Juneja
Dec 14 '18 at 10:48
1
$begingroup$
I'll assume $a_1$ and $b_1$ are the smallest elements of $A$ and $B$. If you have some wiring solution where they are not connected, then $a_1$ connects to $b_u$ and $a_v$ connects to $b_1$ for some $u$ and $v$. Show that if you swap the connections to $a_1-b_1$ and $a_v-b_u$ then the total length of wire cannot increase. So to get an optimal solution, you can assume without loss of generality that $a_1$ and $b_1$ are connected. Then induction takes care the rest, showing that the proposed solution is optimal. It is not always the only optimal solution, but is the only general solution.
$endgroup$
– Jaap Scherphuis
Dec 14 '18 at 14:47
$begingroup$
@JaapScherphuis thank you very mcuh
$endgroup$
– Kavita Juneja
Dec 14 '18 at 15:25
1
1
$begingroup$
How do you connect two real numbers "via lengths of wire?" Are you asking for some permutation $piin S_n$ such that the summation $sumlimits_{i=1}^n |a_i - b_{pi(i)}|$ is minimized?
$endgroup$
– JMoravitz
Dec 14 '18 at 7:07
$begingroup$
How do you connect two real numbers "via lengths of wire?" Are you asking for some permutation $piin S_n$ such that the summation $sumlimits_{i=1}^n |a_i - b_{pi(i)}|$ is minimized?
$endgroup$
– JMoravitz
Dec 14 '18 at 7:07
$begingroup$
@ShubhamJohri: Notice that the sets have the same number of elements.
$endgroup$
– David G. Stork
Dec 14 '18 at 7:56
$begingroup$
@ShubhamJohri: Notice that the sets have the same number of elements.
$endgroup$
– David G. Stork
Dec 14 '18 at 7:56
$begingroup$
@JMoravitz yes, exactly.. you are right. :)
$endgroup$
– Kavita Juneja
Dec 14 '18 at 10:48
$begingroup$
@JMoravitz yes, exactly.. you are right. :)
$endgroup$
– Kavita Juneja
Dec 14 '18 at 10:48
1
1
$begingroup$
I'll assume $a_1$ and $b_1$ are the smallest elements of $A$ and $B$. If you have some wiring solution where they are not connected, then $a_1$ connects to $b_u$ and $a_v$ connects to $b_1$ for some $u$ and $v$. Show that if you swap the connections to $a_1-b_1$ and $a_v-b_u$ then the total length of wire cannot increase. So to get an optimal solution, you can assume without loss of generality that $a_1$ and $b_1$ are connected. Then induction takes care the rest, showing that the proposed solution is optimal. It is not always the only optimal solution, but is the only general solution.
$endgroup$
– Jaap Scherphuis
Dec 14 '18 at 14:47
$begingroup$
I'll assume $a_1$ and $b_1$ are the smallest elements of $A$ and $B$. If you have some wiring solution where they are not connected, then $a_1$ connects to $b_u$ and $a_v$ connects to $b_1$ for some $u$ and $v$. Show that if you swap the connections to $a_1-b_1$ and $a_v-b_u$ then the total length of wire cannot increase. So to get an optimal solution, you can assume without loss of generality that $a_1$ and $b_1$ are connected. Then induction takes care the rest, showing that the proposed solution is optimal. It is not always the only optimal solution, but is the only general solution.
$endgroup$
– Jaap Scherphuis
Dec 14 '18 at 14:47
$begingroup$
@JaapScherphuis thank you very mcuh
$endgroup$
– Kavita Juneja
Dec 14 '18 at 15:25
$begingroup$
@JaapScherphuis thank you very mcuh
$endgroup$
– Kavita Juneja
Dec 14 '18 at 15:25
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039049%2fshortest-length-of-wire-required-to-connect-points%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039049%2fshortest-length-of-wire-required-to-connect-points%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
How do you connect two real numbers "via lengths of wire?" Are you asking for some permutation $piin S_n$ such that the summation $sumlimits_{i=1}^n |a_i - b_{pi(i)}|$ is minimized?
$endgroup$
– JMoravitz
Dec 14 '18 at 7:07
$begingroup$
@ShubhamJohri: Notice that the sets have the same number of elements.
$endgroup$
– David G. Stork
Dec 14 '18 at 7:56
$begingroup$
@JMoravitz yes, exactly.. you are right. :)
$endgroup$
– Kavita Juneja
Dec 14 '18 at 10:48
1
$begingroup$
I'll assume $a_1$ and $b_1$ are the smallest elements of $A$ and $B$. If you have some wiring solution where they are not connected, then $a_1$ connects to $b_u$ and $a_v$ connects to $b_1$ for some $u$ and $v$. Show that if you swap the connections to $a_1-b_1$ and $a_v-b_u$ then the total length of wire cannot increase. So to get an optimal solution, you can assume without loss of generality that $a_1$ and $b_1$ are connected. Then induction takes care the rest, showing that the proposed solution is optimal. It is not always the only optimal solution, but is the only general solution.
$endgroup$
– Jaap Scherphuis
Dec 14 '18 at 14:47
$begingroup$
@JaapScherphuis thank you very mcuh
$endgroup$
– Kavita Juneja
Dec 14 '18 at 15:25