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From: shymaa arafat <shymaa.arafat@gmail.com>
Date: Thu, 16 Sep 2021 17:05:24 +0200
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To: Vincent <vincent.palazzo@protonmail.com>
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Cc: Bitcoin Protocol Discussion <bitcoin-dev@lists.linuxfoundation.org>,
 lightning-dev <lightning-dev@lists.linuxfoundation.org>
Subject: Re: [bitcoin-dev] [Lightning-dev] Storing the Merkle Tree in a
	compact way
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It could be viewed as the simple complete tree to 1D array  with no
pointers described in lecture 8 here
https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-0=
06-introduction-to-algorithms-spring-2020/lecture-notes/index.htm
starting from min 15 in this video
https://youtu.be/Xnpo1atN-Iw

Since all trees in Utreexo forest are full binary trees, this is perfect to
use, and we can save *76*10=E2=81=B6*2*size of pointer(probably4bytes)*
*~600MB *with almost no effort.

However, I suggest to put it in a 2D array to make it more easy to handle
(the indexing math) as we, different than the lecture, traverse in many
ways ( normally to delete or insert, and the parent siblings for the proofs=
)

I wrote more details here
https://bitcointalk.org/index.php?topic=3D5360009.0

On Thu, Sep 16, 2021, 14:37 Vincent <vincent.palazzo@protonmail.com> wrote:

> Hi.
>
> Thanks for the reference, but I missed where you want save space with thi=
s
> compression on the Merkle Tree.
>
> Regards.
>
> Vincent.
> vincenzo.palazzo@protonmail.com
> https://github.com/vincenzopalazzo
> =E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90 Original =
Message =E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90
> On Thursday, September 16th, 2021 at 5:15 AM, shymaa arafat <
> shymaa.arafat@gmail.com> wrote:
>
> Allow me to introduce this simple idea that could be useful ...
>
> -The Intuition was some discussion on Utreexo project about storage savin=
g
> and some traversing issues in handling the UTXOS Merkle Tree/ forest; tha=
t
> is  N internal nodes need to be stored along with 2N pointers (left&right=
),
> + maybe 1 more pointer in the leaves special nodes to handle different
> traversing options (insert, delete, & differently proof fetch that traver=
se
> aunt or niece node according to your implementation
> https://github.com/mit-dci/utreexo/discussions/316)
> .
> Then, I thought of a simple idea that gets rid of all the pointers;
> specially appealing when we have all trees are full (complete) in the
> forest, but can work for any Merkle Tree:
>
> - 2D array with variable row size; R[j] is of length (N/2^j)
> -For example when N=3D8 nodes
> R[0]=3D0,1,2,...,7
> R[1]=3D8,9,10,11
> R[2]=3D12,13
> R[3]=3D14
> .
> -We can see that total storage is just 2N-1 nodes,
> no need for pointers, and traversing could be neat in any direction with
> the right formula:
>
> -Pseudo code to fetch proof[i] ...
>
> //direction to know + or -
> If ((i mod 2)=3D=3D0) drct=3D1;
>             else drct=3D-1;
> // first, the sibling node
> proof[i]=3DR[0,i+drct]
>
> //add the rest thru loop
> For(j=3D1; j=E2=89=A4logN; j++)
>  { index=3D i/(2^j)+drct;
>     proof[i]=3DAdd(R[j,index]);
>  }
>
> -In fact it's just the simple primitive approach of transforming a
> recursion to an iteration, and even if Utreexo team solved their problem
> differently I thought it is worth telling as it can work for any Merkle T=
ree
> .
> Thanks for your time,
> Shymaa M Arafat
>
>
>

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Content-Type: text/html; charset="UTF-8"
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<div dir=3D"auto"><div>It could be viewed as the simple complete tree to 1D=
 array=C2=A0 with no pointers described in lecture 8 here</div><div dir=3D"=
auto"><a href=3D"https://ocw.mit.edu/courses/electrical-engineering-and-com=
puter-science/6-006-introduction-to-algorithms-spring-2020/lecture-notes/in=
dex.htm" rel=3D"noreferrer noreferrer noreferrer" target=3D"_blank">https:/=
/ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-006-intr=
oduction-to-algorithms-spring-2020/lecture-notes/index.htm</a><br></div><di=
v dir=3D"auto">starting from min 15 in this video</div><div dir=3D"auto"><a=
 href=3D"https://youtu.be/Xnpo1atN-Iw" target=3D"_blank" rel=3D"noreferrer"=
>https://youtu.be/Xnpo1atN-Iw</a></div><div dir=3D"auto"><br></div><div dir=
=3D"auto">Since all trees in Utreexo forest are full binary trees, this is =
perfect to use, and we can save <i>76*10=E2=81=B6*2*size of pointer(probabl=
y4bytes)</i></div><div dir=3D"auto"><i>~<b>600MB</b>=C2=A0</i>with almost n=
o effort.</div><div dir=3D"auto"><br></div><div dir=3D"auto">However, I sug=
gest to put it in a 2D array to make it more easy to handle (the indexing m=
ath) as we, different than the lecture, traverse in many ways ( normally to=
 delete or insert, and the parent siblings for the proofs)</div><div dir=3D=
"auto"><br></div><div dir=3D"auto">I wrote more details here</div><div dir=
=3D"auto"><a href=3D"https://bitcointalk.org/index.php?topic=3D5360009.0" r=
el=3D"noreferrer noreferrer noreferrer" target=3D"_blank">https://bitcointa=
lk.org/index.php?topic=3D5360009.0</a><br></div><div dir=3D"auto"><br><div =
class=3D"gmail_quote" dir=3D"auto"><div dir=3D"ltr" class=3D"gmail_attr">On=
 Thu, Sep 16, 2021, 14:37 Vincent &lt;<a href=3D"mailto:vincent.palazzo@pro=
tonmail.com" rel=3D"noreferrer noreferrer noreferrer" target=3D"_blank">vin=
cent.palazzo@protonmail.com</a>&gt; wrote:<br></div><blockquote class=3D"gm=
ail_quote" style=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-le=
ft:1ex"><div>Hi.<br></div><div><br></div><div>Thanks for the reference, but=
 I missed where you want save space with this compression on the Merkle Tre=
e.<br></div><div><br></div><div>Regards.<br></div><div><br></div><div>Vince=
nt.<br></div><div><a href=3D"mailto:vincenzo.palazzo@protonmail.com" rel=3D=
"noreferrer noreferrer noreferrer noreferrer" target=3D"_blank">vincenzo.pa=
lazzo@protonmail.com</a><br></div><div><a href=3D"https://github.com/vincen=
zopalazzo" rel=3D"noreferrer noreferrer noreferrer noreferrer" target=3D"_b=
lank">https://github.com/vincenzopalazzo</a><br></div><div>
        =E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90 Ori=
ginal Message =E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=
=90<br>
        On Thursday, September 16th, 2021 at 5:15 AM, shymaa arafat &lt;<a =
href=3D"mailto:shymaa.arafat@gmail.com" rel=3D"noreferrer noreferrer norefe=
rrer noreferrer" target=3D"_blank">shymaa.arafat@gmail.com</a>&gt; wrote:<b=
r>
        <blockquote type=3D"cite">
            <div dir=3D"auto"><div dir=3D"auto">Allow me to introduce this =
simple idea that could be useful ...</div><div dir=3D"auto"><br></div>-The =
Intuition was some discussion on Utreexo project about storage saving and s=
ome traversing issues in handling the UTXOS Merkle Tree/ forest; that is=C2=
=A0 N internal nodes need to be stored along with 2N pointers (left&amp;rig=
ht), + maybe 1 more pointer in the leaves special nodes to handle different=
 traversing options (insert, delete, &amp; differently proof fetch that tra=
verse aunt or niece node according to your implementation<div dir=3D"auto">=
<a href=3D"https://github.com/mit-dci/utreexo/discussions/316" rel=3D"noref=
errer nofollow noopener noreferrer noreferrer noreferrer noreferrer" target=
=3D"_blank">https://github.com/mit-dci/utreexo/discussions/316</a>)<div dir=
=3D"auto">.<br><div dir=3D"auto">Then, I thought of a simple idea that gets=
 rid of all the pointers; specially appealing when we have all trees are fu=
ll (complete) in the forest, but can work for any Merkle Tree:</div><div di=
r=3D"auto"><br></div><div dir=3D"auto">- 2D array with variable row size; R=
[j] is of length (N/2^j)</div><div dir=3D"auto">-For example when N=3D8 nod=
es</div><div dir=3D"auto">R[0]=3D0,1,2,...,7</div><div dir=3D"auto">R[1]=3D=
8,9,10,11</div><div dir=3D"auto">R[2]=3D12,13</div><div dir=3D"auto">R[3]=
=3D14</div><div dir=3D"auto">.</div><div dir=3D"auto">-We can see that tota=
l storage is just 2N-1 nodes,</div><div dir=3D"auto">no need for pointers, =
and traversing could be neat in any direction with the right formula:</div>=
<div dir=3D"auto"><br></div><div dir=3D"auto">-Pseudo code to fetch proof[i=
] ...</div><div dir=3D"auto"><br></div><div dir=3D"auto">//direction to kno=
w + or -</div><div dir=3D"auto">If ((i mod 2)=3D=3D0) drct=3D1;=C2=A0</div>=
<div dir=3D"auto">=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 else drct=3D-1;=
</div><div dir=3D"auto">// first, the sibling node</div><div dir=3D"auto">p=
roof[i]=3DR[0,i+drct]</div><div dir=3D"auto">=C2=A0 =C2=A0 =C2=A0</div><div=
 dir=3D"auto">//add the rest thru loop</div><div dir=3D"auto">For(j=3D1; j=
=E2=89=A4logN; j++)</div><div dir=3D"auto">=C2=A0{ index=3D i/(2^j)+drct;</=
div><div dir=3D"auto">=C2=A0 =C2=A0 proof[i]=3DAdd(R[j,index]);</div><div d=
ir=3D"auto">=C2=A0}</div><div dir=3D"auto"><br></div><div dir=3D"auto">-In =
fact it&#39;s just the simple primitive approach of transforming a recursio=
n to an iteration, and even if Utreexo team solved their problem differentl=
y I thought it is worth telling as it can work for any Merkle Tree</div><di=
v dir=3D"auto">.</div><div dir=3D"auto">Thanks for your time,</div><div dir=
=3D"auto">Shymaa M Arafat</div><div dir=3D"auto"><br></div></div></div></di=
v>

        </blockquote><br>
    </div></blockquote></div></div></div>

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