VECTOR QUANTIZATION, GRAPH THEORY, AND IMRT DESIGN

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1 VECTOR QUANTIZATION, GRAPH THEORY, AND IMRT DESIGN JOSHUA G. REESE Abstract. Vector quantzaton s an mportant concept n the feld of nformaton theory and codng theory. The p-medan problem s a classc graph theory problem wth natural extensons to operatons research faclty locaton problems. We nvestgate some prevously unrecognzed connectons between these problems, and ultmately show that the problem of desgnng an optmal vector quantzer on a dscrete set X s closely related to solvng the p-medan problem on a graph G = (X, E). We also demonstrate how vector quantzaton and ts connecton to the p-medan problem have mportant applcatons n radotherapy treatment desgn, specfcally n the IMRT beam selecton problem. 1. Introducton Claude E. Shannon lad the groundwork of vector quantzaton (VQ) by assumng the exstence of a dstorton measure that would quantfy the dfference n qualty between an nput and a coded reproducton. He worked on VQ under the guse of source codng subject to a fdelty crteron [3]. The p-medan problem s a classc graph theory problem wth natural extensons n operatons research to problems such as the faclty locaton problem. Ths problem nvolves locatng p facltes on a graph so that the sum of the weghted dstances from the vertces of the graph to ther nearest faclty s mnmzed [1]. We begn by defnng and descrbng these concepts n detal, and then we note some prevously unrecognzed connectons between them. We then apply VQ, the p-medan problem, and the connectons between them to the problem of desgnng a feasble radotherapy treatment desgn. A crtcal radotherapy desgn queston s how to choose a collecton of angles from whch to emt radaton nto the tumor. It s mportant that these angles satsfy clncal restrctons and constrants on the amount of radaton deposted n dfferent areas of the body. We show that vector quantzaton and the p-medan problem are mportant tools for solvng ths problem, and conclude by ntroducng some new avenues of research that our results present. 2. Vector Quantzaton 2.1. Basc Defntons. The materal n ths secton s treated exhaustvely n [3], and nterested readers are drected to ths text. Vector quantzaton s a process for codng contnuous or dscrete vector sgnals subject to a fdelty crteron. It s often used to compress mages or other data. Date: Aprl 27, Key words and phrases. Graph Theory, p-medan, Vector Quantzaton, Radotherapy. Specal thanks to Dr. Allen Holder, Department of Mathematcs, Trnty Unversty. 1

2 2 JOSHUA G. REESE A vector quantzer Q maps vectors from an nput set of vectors X nto a fnte set C (called the codebook) contanng N output ponts (called codevectors or codewords). Thus, Q : X C, where C = {y 1, y 2,..., y N } and y X for each J = {1, 2,..., N}. The vector quantzer separates X nto N dstnct regons called cells, denoted X for J, and defned as X = {x X Q(x) = y }. These cells are dsjont and together nclude all elements of the nput set. Thus we have, X = X and X Xj = for j. So, these ndvdual cells form a partton of X, defned as P Q = {X J }. The basc operaton of a vector quantzer s to partton an nput set of vectors nto partton cells and select a codevector from each cell to serve as a representatve for all of the vectors wthn that cell. Thus, vector quantzers may be separated nto two separate processes, known as the encoder E and decoder D. The encoder s job s to examne each nput vector and to determne whch partton cell t should belong to. Thus, the encoder maps elements from X nto the set J, whch contans ndces of the cells of X. The decoder then chooses a vector from each numbered cell to serve as the codevector for that regon. Thus, the decoder maps elements from J nto the codebook C. We have, E : X J and D : J C. As an example, let X be a fnte set of ponts on a cty map. These ponts could be houses, schools, busnesses, or any other locaton wthn the cty. Consder an encoder that maps chldren s houses nto partton cells, or school dstrcts, each represented by a school dstrct number. These school dstrcts are dsjont subsets of X. The decoder then maps the ndex number of a partcular school dstrct to a locaton wthn that dstrct representng the locaton of the school. Dependng on the desgn of the decoder, the quantzer mght choose a locaton that already has a house, school or busness bult on t, or t could choose an unused locaton Representng VQ Wth Bnary Selector Functons. We represent the operaton of the encoder n terms of selector functons S (x), defned as bnary functons that ndcate whether a vector x s n a gven cell X. Thus, { 1 f x X S (x) = 0 otherwse. Snce the decoder creates a codebook by selectng a codevector from each cell X, t s possble to create a selector functon that descrbes the entre operaton of the quantzer, as follows: { 1 f xj s quantzed as x λ j = 0 otherwse. Notce that the codebook s: C = {x λ = 1}.

3 VECTOR QUANTIZATION, GRAPH THEORY, AND IMRT DESIGN Vector Quantzer Performance. We quantfy the nonnegatve cost of quantzng any nput vector x X wth the codevector Q(x) by measurng dstorton, denoted by d(x, Q(x)). The most common measurement of dstorton between vectors s the Eucldean dstance, d(x, Q(x)) = x Q(x). The methods of quantfyng dstorton are numerous, however, and often depend on the partcular applcaton of VQ. For now we wll smply assume that d(x, Q(x)) s a dstorton metrc meanngful to ts applcaton. We use ths dstorton measure to determne the overall performance of the vector quantzer. Vector quantzaton s nherently a statstcal problem. One of ts orgnal uses was analog to dgtal sgnal converson. For ths reason, vector quantzers were, and stll are, desgned so that even though the nput sgnal s not prevously known, a sutable measure of expected dstorton may be obtaned. In order to work towards ths measure of expected dstorton, we frst state the followng defnton. Defnton 2.1. A functon f X defned on X s called a jont probablty densty functon (pdf) f f X (x) 0, x X, x X f X (x)dx = 1, where the ntegral s taken to be a multple ntegral over the k-dmensonal space. In the school dstrct example, f X corresponds to the set of ponts on a map, f X (x) s the probablty that a locaton x s a house wth school-age chldren. There are many dfferent factors that may determne probablty, dependng on the partcular applcaton of VQ. If we model the quantzer as a random process, ts performance corresponds to ts expected dstorton, whch by defnton s D Q = Ed(x, Q(x)) = d(x, Q(x))f X (x)dx, where f X (x) s a jont probablty densty functon (pdf) of the nput vector space X. Here we are evaluatng a statstcal average of the dstorton measure over the entre nput vector space. Ths measure s called the average dstorton of the vector quantzer. If the nput set of vectors has a dscrete dstrbuton (as opposed to a contnuous dstrbuton), then we have D Q = Ed(x, Q(x)) = f X (x )d(x, Q(x )), where each x s a value n X that has nonzero probablty. In terms of the selector functon descrbed n Secton 2.2, D Q = f X (x j )d(x, x j )λ j, (2.1) j where λ j = 1 f and only f Q(x j ) = x.

4 4 JOSHUA G. REESE 2.4. Optmal Vector Quantzer Desgn. The vector quantzer desgn problem s: Fnd a codebook C specfyng the decoder D and a partton P Q specfyng the encoder E such that the average dstorton D Q s mnmzed. If we fnd such a codebook and partton, then the resultng encoder E and decoder D produce a globally optmal quantzer. We now state three condtons that lead to algorthms for codebook mprovement. These condtons are often called the necessary condtons for optmal vector quantzers. See [3] for more detals and proofs of the followng condtons. (1) Partton Condton. Gven a codebook C, a partton s optmal f and only f t assgns vectors to the codevectors y n a nearest neghbor fashon,.e., Q(x) = y d(x, y ) d(x, y j ) j J. In other words, gven a fxed codebook, t s both necessary and suffcent for optmalty that the partton satsfy the Partton Condton. (2) Codebook Condton. Gven a partton P Q, a codebook s optmal f and only f t mnmzes the dstorton between vectors x X and the codevectors y, averaged over the probablty dstrbuton of x gven that x les n X. Hence, E[d(x, y ) x X ] E[d(x, y) x X ] for all y X. Ths condton s farly ntutve, as t ensures that each codevector mnmzes the expected dstorton over ts respectve partton cell. Ths condton s both necessary and suffcent for producng an optmal codebook, gven a fxed partton. (3) Boundary Condton. Let B = {x d(x, y ) = d(x, y j ) for some j} be the boundary ponts of the gven partton. These ponts are equdstant from two codevectors and hence do not have a unque nearest neghbor. An optmal codebook for a gven source dstrbuton must assgn these ponts to one and only one partton cell. In order to practcally ensure that ths condton holds, an effectve te-breakng scheme must be mplemented when desgnng the encoder. Returnng to the school dstrct example, the frst condton ensures that an optmal partton, or school dstrct plan, s one n whch the houses are closer to that dstrct s school than any other school. Lkewse, the second condton guarantees that an optmal set of school locatons gven a set of school dstrcts s one n whch the schools mnmze dstorton, averaged over the probablty dstrbuton, between the school and each house n a gven dstrct. The thrd condton ensures that no house les wthn two school dstrcts at the same tme Lloyd Algorthm. The Lloyd algorthm s useful for desgnng an effectve quantzer. It s based on the fact that the Partton Condton and Codebook Condton allow us to obtan an optmal codebook gven a fxed partton, and an optmal partton gven a fxed codebook. The algorthm requres an ntal, dscrete set of vectors called the tranng set. If the set we are tryng to quantze s dscrete, then we may smply apply the Lloyd Algorthm to ths set. If t s contnuous, then we select a set of observatons from t and use these as the tranng set for the Lloyd Algorthm. The algorthm begns wth an ntal codebook, and at every teraton

5 VECTOR QUANTIZATION, GRAPH THEORY, AND IMRT DESIGN 5 produces a new codebook that s guaranteed to ether reduce or leave unchanged the overall dstorton D. The algorthm s a three step process, as follows: Lloyd Algorthm (1) Set m = 1. Begn wth an ntal codebook C 1. (2) (The Lloyd Iteraton) Gven the codebook C m, create partton cells by usng the Partton Condton as follows: R = {v d(v, y ) < d(v, y j ), j }. If d(v, y ) = d(v, y j ) for any j, then break the te by assgnng v to the cell R j for whch j s smallest, thus satsfyng the Boundary Condton. Usng the Codebook Condton, we create a new codebook C m+1, whch s optmal for the partton P m = {R 1 N}. (3) Compute D, the average dstorton for C m+1, and f t has decreased less than δ, stop. Otherwse, set m + 1 m and go to step 2. Theorem 2.2 (Gersho and Gray [3]). Each applcaton of the Lloyd Iteraton reduces or leaves unchanged the average dstorton. Proof. From the necessary condtons for optmalty, the Partton Condton mples that the frst part of the Lloyd Iteraton can only mprove or leave unchanged the encoder for the gven decoder. Smlarly, the Codebook Condton mples that the second part can only mprove or leave unchanged the decoder for the gven encoder. Hence, the average dstorton of the vector quantzer cannot ncrease. Theorem 2.3 (Gersho and Gray [3]). For a fnte tranng set, the Lloyd Algorthm always produces a sequence of vector quantzers whose average dstortons converges n a fnte number of teratons. Proof. There are a fnte number of ways to partton the nput set nto N subsets. For each partton, the Codebook Condton yelds a codebook wth a partcular value of average dstorton. The monotone nonncreasng average dstorton mples that the algorthm cannot return n subsequent teratons to a partton yeldng a hgher average dstorton. Hence, the average dstorton of the sequence of vector quantzers produced must converge to a fxed value n a fnte number of steps. 3. Graph Theory and Desgnng Optmal VQ Codebooks 3.1. Introducton. The p-medan problem has long been useful n solvng operatons research problems dealng wth faclty locaton. The problem s to choose a p-element subset of a graph such that the weghted sum of the dstances between the vertces of the graph and ther nearest element n the subset s mnmzed [1] P-Medan Problem. Let G = (X, E) be a complete, weghted and undrected graph, where X s the set of vertces and E s the set of edges. Let d j = [d(x, x j )] be the shortest dstance matrx where the entry n the th row and jth column contans the shortest dstance between vertces x and x j, accordng to some metrc

6 6 JOSHUA G. REESE d. Let w(x ) be the vertex weghts. In the faclty locaton problem, these are the costs of satsfyng the demand at each vertex x. Let ξ j be an allocaton varable such that { 1 f vertex xj s allocated to vertex x ξ j = 0 otherwse. The nteger optmzaton formulaton of the p-medan problem s: mn w(x j )d j ξ j j subject to: ξ j = 1 for j = 1,..., n ξ j ξ ξ = p, j = 1,..., n ξ j {0, 1}. (3.1) The constrant ξ j = 1 for j = 1,..., n ensures that each vertex s allocated to one and only one element n the p-element subset. The second constrant, ξ = p, guarantees that there are p vertces allocated to themselves, whch forces the cardnalty of the p-medan subset to be p. The last two constrants ensure that the allocaton varables are ether 0 or 1. If ξj s an optmal answer to the problem defned by (3.1), then the p-medans are the elements of: X p = {x ξ = 1} Usng the P-Medan Problem to Desgn Optmal VQ Codebooks. The followng theorem shows an mportant connecton between VQ and the p- medan problem n that we may desgn an optmal VQ codebook by solvng a partcular p-medan problem for a fxed p > 0. Theorem 3.1. Let X be a fnte set of vectors. Let f X (x) be a pdf of X, and d(x, x j ) be a metrc on X. An optmal vector quantzer Q : X C, C = p, may be found by solvng the p-medan problem on the graph G = (X, E) wth vertex weghts f X (x), obtanng an optmal soluton X p = C, where C s the optmal codebook for Q. Proof. Let w(x j ) = f X (x j ), j, and let d j = [d(x, x j )]. Then by solvng (3.1), we obtan [ξ j ] such that f X (x j )d(x, x j )ξ j j s mnmzed. The constrant ξ j = 1 for j = 1,..., n corresponds to the Boundary Condton for vector quantzers and ensures that each vertex s allocated to one and only one element n the p-element subset (codebook). The constrant ξ = p ensures that the number of outputs s p,.e. X p = C = p. The last two constrants guarantee that the allocaton varables, whch correspond to the selector functons λ j n VQ, are ether 0 or 1. Then, f λ j = ξ j, we have that the average dstorton of the quantzer gven by (2.1) s mnmzed. We conclude that C = {x λ = 1} = X p = {x ξ = 1} s an optmal codebook for Q.

7 VECTOR QUANTIZATION, GRAPH THEORY, AND IMRT DESIGN Soluton Technques. Karv and Hakm [6] showed that the problem of fndng a p-medan on a graph s NP-hard n p and n, even when the graph s planar, has maxmum vertex degree 3, and all of the edges have length 1 and all vertex weghts are 1. Polynomal algorthms exst, however, for the case when p s fxed. Theorem 3.2. Let n = X. The worst-case complexty of the p-medan problem on G = (X, E) for a fxed p N s O(pn p+1 ). Proof. Before the algorthm begns, a shortest path dstance matrx s created usng Djkestra s algorthm, contanng the shortest weghted dstances between all vertces of the graph. The complexty of ths step s O(n 2 ) [1]. The search then proceeds through every p-element subset of X, requrng ( ) n n p p calculatons. For each p-element subset of X, we must compare every x X to the p elements of the subset, to satsfy the nearest neghbor condton. Ths requres pn calculatons. So, the overall complexty s no greater than O(n 2 + (pn)n p ) = O(pn p+1 ). Gven Theorems 3.1 and 3.2, the followng corollary becomes apparent: Corollary 3.3. The problem of desgnng an optmal quantzer on a dscrete set of data s polynomal. 4. Applcatons n Radotherapy Treatment Desgn 4.1. Introducton. Intensty Modulated Radotherapy Treatment (IMRT) s a method of cancer treatment that drects onzng beams of radaton through the anatomy and nto a tumorous regon. The goals of IMRT are to 1) drect a unform amount of radaton nto the tumor and 2) lmt the amount of radaton that deposts n nearby crtcal and normal healthy tssues. The tumor should receve enough radaton to kll the cancer cells, and the crtcal structures should receve no more than a prevously specfed amount of radaton n order to reduce damage [4]. The radaton deposted per unt mass of tssue s called absorbed dose and s measured n Gray (Gy), whch s equal to one Joule of energy deposted per klogram [5] Theraputc Advantage. Cancer cells are nnately focused on reproducng and lack the effectve capablty to repar themselves. Consequently, cancer cells are klled when ther DNA s damaged, whle healthy cells are able to contnue reproducng after mnor damage by rradaton [2]. The dfference between the radaton tolerance of cancer and normal cells s called the therapeutc advantage. IMRT treatments take advantage of ths therapeutc advantage by delverng the full dose prescrbed to the tumor n smaller daly doses [5]. Ths method of treatment klls the tumor cells, but gves enough tme between doses for the normal tssue to heal. It also allows the dead cancer cells to be carred away by the healthy tssue around the tumor, preventng a collecton of dead tssue where the tumor was located.

8 8 JOSHUA G. REESE 4.3. IMRT Treatment Method. Before the treatment takes place, physcans use a seres of sequental CT scans to obtan a 3D representaton of the tumor and any nearby crtcal structures. Physcans then ndcate on each CT scan the locaton of these structures, and the remanng tssue s labeled as normal tssue. The prescrpton s then specfed by the followng varables [4]: T LB The lower bound for absorbed dose n the tumor. T UB The upper bound for absorbed dose n the tumor. CUB The upper bound for absorbed dose n the crtcal structures. NUB The upper bound for absorbed dose n the normal tssue. Each crtcal structure s assgned a dfferent upper bound that vares wth the structure type and s determned by the susceptblty and severty of damage to that structure. For nstance, a sgnfcant amount of damage to the spnal cord may result n paralyss or loss of crtcal body functons. Therefore, the tolerance assgned to the spnal cord s comparatvely lower than most other structures [5]. After the physcans determne an IMRT plan that meets the constrants of the prescrpton, the patent s lad on a couch and mmoblzed. The devce that emts the radaton s called a lnear accelerator, or LINAC. The LINAC s mounted on a movable arm, called the gantry, that s able to make a complete 360 degree rotaton around the patent. Ths allows physcans to delver varyng radaton doses from any partcular collecton of angles. The gantry rotates around a pont called the socenter, and patents are postoned so that the socenter s located n the mddle of the tumor. LINACs are usually equpped wth a multleaf collmator, a devce wth pneumatc leaves that move back and forth to block portons of the radaton beam. The multleaf collmator allows the beam to be shaped, thereby parttonng each of the 360 angles nto sub-beams, also known as pencl beams [5] Optmal IMRT Plan Desgn. The followng s a bref dscusson of the constructon of an IMRT plan. See [2], [4] and [5] for further detals. There are three separate problems to solve when desgnng an IMRT plan. They are 1) determnng whch angles should be used to focus radaton on the patent, 2) fndng a fluency pattern (a collecton of exposure tmes) correspondng to these angles, and 3) choosng a delvery sequence that effcently admnsters ths treatment [2]. Ths paper nvestgates the frst problem, whch s known as the beam selecton problem. Lnear programs are often used to determne optmal treatments, consstng of fluency values, or radaton exposure tmes, for each angle and sub-beam. Together these values defne the fluency vector x, where x (a,) s the exposure tme for the th sub-beam of angle a, for a A = {π/180 o = 0, 1, 2,..., 359}. The dose matrx A s an approxmaton of how radaton s deposted nto each dose pont n the anatomy. The unts of A are Gy per unt tme. The lnear transformaton x Ax maps the fluency vector nto the anatomy and determnes the amount of radaton deposted at every dose pont n the anatomy [2]. The lnearty s not just an assumpton of the model t s expermentally valdated that the dose s deposted lnearly [5]. We partton A nto full-column sub-matrces A T, A C and A N, contanng the dose ponts correspondng to the tumor, crtcal structures and normal tssue, respectvely. There are several lnear programs n the lterature for producng optmal IMRT plans, and the followng s an example:

9 VECTOR QUANTIZATION, GRAPH THEORY, AND IMRT DESIGN 9 mn{ω l T α + u T Cβ + u T Nγ} subject to: T LB Lα A T x T UB A C x CUB + U C β A N x NUB + U N γ 0 Lα T LB CUB U C β 0 U N γ 0 x, where ω s a weght determnng the mportance of meetng the lower bound on the tumor. The frst three constrants are elastc and are allowed to vary wth α, β and γ, respectvely. These vectors represent how much the model s allowed to volate the radaton constrants on the tumor, crtcal structures and normal tssue, respectvely. The matrces L, U C, and U N defne how elastcty s measured, and each corresponds wth a respectve vector, l, u C and u N that decdes how dscrepances are penalzed. The decson values are x, α, β and γ [4]. Ths model has been mplemented n a prototype treatment package named Radotherapy OptmAl Desgn (RAD). RAD allows users to delneate a tumor, crtcal structures, and normal tssue on a 64 x 64 grd (smlar to how a physcan would ndcate these structures on a CT scan). One can then assgn prescrpton values to these structures and solve for the optmal plan correspondng to that problem nstance [4] IMRT Beam Selecton Problem. Unfortunately, delverng a treatment from all 360 gantry angles s not clncally feasble. Current clncal constrants requre that IMRT treatments consst of 7-9 gantry angles from whch to delver radaton. Ths s prmarly because physcans lmt the treatment tme n order to reduce patent movement, whch may result n treatment error. Thus, the beam selecton problem s how to select p angles so that the subsequent p-angle treatment s effectve. Solvng nteger programs desgned to choose p angles exceeds calculaton capabltes. An alternatve strategy s to begn wth an optmal treatment lke the one descrbed above and strategcally prune angles to form an effectve and clncally feasble treatment. In the next secton we dscuss how an optmal vector quantzer addresses ths problem. 5. Optmal VQ and IMRT Treatment Desgn 5.1. Characterstc Vectors. We defne a k-dmensonal characterstc vector c a for a gven angle a as c a = (g 1 (a), g 2 (a),..., g k (a)), where g : A R, A = {π/180 o = 0, 1, 2,..., 359}. The components g (a) are dfferent statstcs or characterstcs of the angle a as t s used n the optmal 360 degree plan. We measure c a by a norm c a, and ths norm nduces the metrc: d(a, a j )) = c a c aj. Usng the characterstc vectors allows us to measure an angle s effectveness wth respect to a wde varety of user-defned statstcs, ncludng statstcs on what the apparent effect each angle has on the qualty of treatment. Angle locaton s perhaps the most obvous characterstc of a gven angle. Prelmnary efforts at ncorporatng

10 10 JOSHUA G. REESE VQ and IMRT have met wth some success n usng a pure angle dstance metrc [2]. Ths yelds a one-dmensonal characterstc vector, whch contans angle locaton,.e. c a = (πa/180 o ). The method presented here allows us to ncorporate numerous other factors, ncludng how each angle contrbutes to the overall prescrpton and the absorbed dose wthn the anatomy. If we let X be a set of characterstc vectors, then usng VQ on X allows us to 1) group angles accordng to how smlar ther characterstcs are, and 2) to select a codebook of p characterstc vectors to represent X such that the average dstorton of dong so s mnmzed. Therefore, the angles that correspond to these characterstc vectors satsfy the objectve of the beam selecton problem. These angles may or not be truly optmal however, as ther optmalty depends on what statstcal data s present n the characterstc vectors. Hence, ths technque s smply a heurstc to solvng the overall problem Vector Quantzaton and the Beam Selecton Problem. We frst let f(a ) be a pdf that descrbes the statstcal characterstcs of the angles n the optmal 360 angle plan. Ultmately the process presented here s an attempt to solve a determnstc optmzaton problem va a statstcal technque. Ths fact places ths technque squarely n the realm of probablty modellng. The pdf should contan nformaton regardng the lkelhood of usng any gven angle. It s an open queston as to how to create an effectve pdf for ths problem, although there are several theores. One example s to calculate how much absorbed dose each ndvdual angle can delver to the tumor before volatng the constrants on the crtcal structures or normal tssue. Here we are measurng each angle s potental for treatng the tumor, and the assumpton s that the potental for a gven angle translates nto the probablty of that angle beng used. Untl further results are gathered we wll assume that f(a ) s an approprate pdf for ths partcular applcaton. Once we have an approprate pdf, we defne a quantzer Q : X C, where C = p, and C X. If we let w(c a ) = f(a ),, then by Theorem (3.1) we can fnd the optmal codebook C by solvng the p-medan problem on the graph G = (X, E). Furthermore, we have that ths codebook contans the characterstc vectors assocated wth the angles that are solutons to the beam selecton problem. Ths collecton of p angles s a soluton to the beam selecton problem n the sense that there exsts no other collecton of p angles that can be selected wth lower average dstorton (based upon the nformaton n the characterstc vector) of the orgnal 360 angle plan. 6. New Drectons Our results suggest new areas of research, and the followng questons should be nvestgated. What angle statstcs or characterstcs should be ncluded n the characterstc vector? How do we best select these characterstcs to accurately depct angle dfferences accordng to what s actually happenng n the anatomy? What s an approprate probablty densty functon for an optmal treatment consstng of 360 angles?

11 VECTOR QUANTIZATION, GRAPH THEORY, AND IMRT DESIGN 11 How may we reduce the complexty of solvng the p-medan problem, and what are ntellgent heurstcs? Can we use the p-medan problem to make statements about the Lloyd Algorthm? References [1] N. Chrstofdes. Graph Theory: An Algorthmc Approach. Academc Press, [2] M. Ehrgott and A. Holder. Beam selecton n radotherapy desgn. Unpublshed Draft, [3] A. Gersho and R. M. Gray. Vector Quantzaton and Sgnal Compresson. Kluwer Internatonal Seres n Engneerng and Computer Scence. Kluwer Academc Publshers, [4] A. Holder. Radotherapy treatment desgn and lnear programmng. In M. Brandeau, F. Sanfort, and W. Perskalla, edtors, Operatons Research and Health Care: A Handbook of Methods and Applcatons, chapter 29. Kluwer Academc Publshers, [5] A. Holder and B. Salter. A tutoral on radaton oncology and optmzaton. In H. Greenberg, edtor, Tutorals on Emergng Methodologes and Applcatons n Operatons Research, chapter 4. Kluwer Academc Publshers, [6] O. Karv and S. Hakm. An algorthmc approach to network locaton problems. : The p-medans. SIAM Journal on Appled Mathematcs, 37(3): , December Current address: One Trnty Place 1817, San Antono, Texas E-mal address: jreese@trnty.edu